UC-NRLF 


•.*••     it 


LIBRARY 

OF  THK 

UNIVERSITY  OF  CALIFORN 


f 

<!X6 


Received 
Accession  No. 


7 


.    Class  No. 


_J 


JOHN   SWEtT. 


PRESENTJ?t>   BY 


ROBINSON'S  MATHEMATICAL  SERIES. 
THE 

RUDIMENTS 

•V-y,.<. 

OP 

WRITTEN  ARITHMETIC: 

CONTAINING 

SLATE  AM)  BLACK-BOARD  EXERCISES  FOR  BEGIMERS, 

AND   DESIGNED    FOR 

GBADED  SCHOOLS. 

EDITED    BY 

W.  FISH,  A.M. 


NEW  YOKK: 
,  PHINNEY,  BLAKEMAN  & 
CHICAGO :   S.  C.  GRIGGS  &  CO. 

1866. 


ROBINSOST  S 


The  most  COMPLETE,  most  PRACTICAL,  and  most  SCIENTIFIC  SERIES 
of  MATHEMATICAL  TEXT-BOOKS  ever  issued  in  this  country. 

(Esr  Tw:m:EsrTY-T~w~o 

+  7J-3* 


Robinson's  Progressive  Table  Book, 

Hobinson's  Progressive  Primary  Arithmetic,  - 
Hobinson's  Progressive  Intellectual  Arithmetic,   - 
Hobinson's  Rudiments  of  "Written  Arithmetic,     - 
.Robinson's  Progressive  Practical  Arithmetic, 
Hobinson's  Key  to  Practical  Arithmetic,  -       -       - 
Hobinson's  Progressive  Higher  Arithmetic,    - 
Robinson's  Key  to  Higher  Arithmetic,    ----- 
Robinson's  Arithmetical  Examples,  - 

Robinson's  New   Elementary  Algebra, 

Robinson's  Key  to  Elementary  Algebra, 

Hobinson's  University  Algebra,   -       -       -       -       - 
Hobinson's  Key  to    University  Algebra,   - 
Hobinson's  New   University  Algebra,        -     *  - 
Hobinson's  Key  to  New  University  Algebra,  - 
Hobinson's  New  Geometry  and  Trigonometry,    - 
Hobinson's  Surveying  and  Navigation,     -       -       -       - 
Hobinson's  Analyt.  Geometry  and  Conic  Sections, 
Hobinson's  Differen.  and  Int.  Calculus,  (in  preparation,)- 

Itobinson's  Elementary  Astronomy, 

Hobinson's  University  Astronomy,     ------ 

Robinson's  Mathematical  Operations,        - 
Hobinson's   Key  to   Geometry  and  Trigonometry,   Conic 
Sections  and  Analytical  Geometry, 


Entered,  according  to  Act  of  Congress,  in  the  year  1861, 
and  again  in  the  year  1S63,  by 

DANIEL    W.    FISH,    A.M., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States,  for  the  Northern 
District  of  New  York. 


PREFACE. 


In  the  preparation  of  this  work,  a  special  object  has  been 
kept  in  view  by  the  author,  namely;  to  furnish  a  small  and 
simple  class  book  for  beginners,  which  shall  contain  no 
more  of  theory  than  is  necessary  for  the  illustration  and 
application  of  the  elementary  principles  of  written  arith- 
metic, applied  to  numerous,  easy,  and  practical  examples, 
and  which  shall  be  introductory  to  a  full  and  complete 
treatise  on  this  subject. 

This  book  is  not  to  be  regarded  as  a  necessary  part  of 
the  Arithmetical  Series  by  the  same  author,  as  the  four 
books  already  composing  that  Series  are  believed  tu  be 
properly  and  scientifically  graded,  and  eminently  adapted 
to  general  use  j  but  this  work  has  been  prepared  to  meet 
a  limited  demand,  in  large  graded  schools,  and  in  the  pub- 
lic schools  of  New  York,  and  similar  cities,  where  a  large 
number  of  pupils  often  obtain  but  a  limited  knowledge  of 
arithmetic,  and  wish  to  commence  its  study  quite  young ; 
and  it  is  also  designed  for  those  who  desire  a  larger  num 
ber  of  simple  and  easy  exercises  for  the  slate  and  black- 
board than  are  usually  found  in  a  complete  work  on  writ- 
ten arithmetic,  so  that  the  beginner  may  acquire  facility, 
promptness,  and  accuracy  in  the  application  and  operations 
of  the  fundamental  principles  of  this  science. 

(iii) 


IV  PREFACE. 

The  principles,  definitions,  rules,  and  applications  so  far 
as  developed  in  this  work  coincide  with  the  other  books 
of  the  same  series.  Many  of  the  Contractions,  and  special 
applications  of  the  rules,  particularly  those  that  are  at  all 
difficult,  have  been  omitted,  and  also  the  treatment  of  De- 
nominate Fractions,  and  Decimals,  all  of  which  are  fully 
and  practically  treated  in  the  Progressive  Practical,  and 
the  Higher  Arithmetic.  -A  few  easy  and  practical  appli- 
cations of  Cancellation,  Analysis,  Per  centage  and  Sim- 
ple Interest  have  been  given,  and  a  very  large  number  ot 
easv  examples. 


CONTENTS. 


SIMPLE   NUMBERS.  Page, 

Definitions,  , 7 

Roman  Notation, 8 

Arabic  Notation, 9 

Laws  and  Rules  for   Notation  and  Numeration, 16 

• 

Addition, ..18 

Subtraction, 29 

Multiplication, ., . . .  39 

Contractions, 48 

Division, 54 

Contractions, 68 

Problems  in  Simple  Integral  Numbers, 72 

COMMON    FRACTIONS. 

Definitions,  Notation  and  Numeration, .. 74 

Reduction  of  Fractions, ~. .  .78 

Addition  of  Fractions, 83 

Subtraction  of  Fractions, 86 

Multiplication  of  Fractions, 88 

Division  of  Fractions, ^^ 94 

DECIMALS. 

• 

Notation  and  Numeration, 102 

Reduction  of  Decimals, 107 

Addition  of  Decimals, 1$) 


VI  CONTENTS. 

Page. 
Subtraction  of  Decimals, Ill 

Multiplication  of  Decimals, 112 

Division  of  Decimals, 114 

UNITED    STATES    MONEY. 

Reduction  of  United  States  Money, 118 

Addition  of  U.  S.  Money, ^ 120 

Subtraction  of  U.  S.  Money, 122 

Multiplication  of  U.  S.  Money, 124 

Division  of  U.  S.  Money, 125 

Bills, ..."....  128 

COMPOUND  NUMBERS. 

Weights  and  Measures, 130 

Aliquot  parts, 145 

Reduction  Descending, 146 

Reduction  Ascending, 148 

Addition  of  Compound  Numbers, '.  153 

Subtraction  of  Compound  Numbers, 156 

Multiplication  of  Compound  Numbers, %. . .  159 

Division  of  Compound  Numbers, 162 

CANCELLATION, 167 

ANALYSIS, 172 

PERCENTAGE, 177 

COMMISSION, 179 

PROFIT  AND  Loss, 180 

INTEREST, „ 181 

PROMISCUOUS  EXAMPLES, 186 


RUDIMENTS  OF  ARITHMETIC. 


DEFINITIONS. 

1 .  Quantity  is  any  thing  that  can  be  increased,  dimin- 
ished, or  measured  ;  as  distance,  space,  weight,  motion,  time, 
-  2.  A  Unit  is  one,  a  single  thing,  or  a  definite  quantity. 

3.  A  Number  is  a  unit,  or  a  collection  of  units. 

4.— 3Vn  Abstract  Number  is  a  number  used  without  ref- 
erence to  any  particular  thing  or  quantity ;  as  3,  ft,  756. 

5~rk:  Concrete  Number  is  a  number  used  with  refer- 
ence to  some  particular  thing  or  quantity;  as  21  ffturs,  4 
cents,  230  miles. 

45.  A  Simple  Number  is  eithej:  an  abstract  nuiriter,  or  a 
concrete  number  of  but  one  denomination;  as  48,  52  pounds, 
36  days. 

7.  A  Compound  Number  is  a  concrete  number  expressed 
in  two  or  more  denominations ;  as  4  bushels  3  pecks,  8  rods 
4  yards  2  feet  3  inches. 

8.  An  Integral  Number,  or  Integer,  is  a  number  which 
expresses  whole  things;  as  5,  12  dollars,  17  men. 

9.  A  Fractional  Number,  or  Fraction,  is  a  number 
which  expresses  equal  parts  of  a  whole  thing  or  quantity; 
as  A,  f  of  a  pound,  75y  of  a  bushel. 

1 0.  Like  Numbers  have  the  same  kind  of  unit,  or  ex- 
press the  same  kind  of  quantity.     Thus,  74  and  16  are  like 
numbers;  so  are  74  pounds,  16  pounds,  and  12  pounds; 
also,  4  weekb  3  days,  and  16  minutes  20  seconds,  both  being 
used  to  express  units  of  time. 


8  SIMPLE   NUMBERS. 

11.  Unlike  Numbers  have  different  kinds  of  units,  or 
are  used  to  express  different  kinds  of  quantity.  Thus,  36 
miles,  and  15  daya ;  5  hours  36  minutes,  and  7  bushels  3 
pecks. 

1£.  Arithmetic  is  the  Science  of  numbers,  and  the  Art 
of  computation. 

13..  The  Five  Fundamental  Operations  of  Arithmetic 
are,  Notation  and  Numeration,  Addition,  Subtraction, 
Multiplication,  and  Division. 

NOTATION  AND  NUMERATION. 

14:J^Notation  is  a  method  of  writing  or  expressing 
numbers  by  characters ;  and, 

l«5j§Numeration  is  a  method  of  leading  numbers  ex- 
pressed by  characters. 

IGf^Two  systems  of  Notation  are  in  general  use — the 
Roman  and  Arabic. 

THE   ROMAN  NOTATION 

M7.  Employs  seven  capital  letters  to  express  numbers, 
thus  : 
Letters,         I        V        X         L        C        D        M 

Values,  one,        five,          ten,          fifty,    hu°nn^ed,  hunted,  thoSLnd. 

•%18.  The  Roman  notation  is  founded  upon  the  following 
principles : 

1st.  Repeating  a  letter  repeats  its  value.  Thus,  II  rep- 
resents two,  XX  twenty,  CCC  three  hundred. 

2d.  If  a  letter  of  any  value  be  placed  after  one  of  greater 
value,  its  value  is  to  be  united  to  that  of  the  greater.  Thus, 
XI  represents  eleven,  LX  sixty,  DC  six  hundred. 

3d.  If  a  letter  of  any  value  be  placed  before  one  of  greater 


NOTATION   AND   NUMERATION.  9 

value,  its  value  is  to  be  taken  from  that  of  the  greater. 
Thus,  IX  represents  nine,  XL  forty,  CD  four  hundred. 

4th.  If  a  letter  of  any  value  be  placed  between  two  letters, 
each  of  greater  value,  its  value  is  to  be  taken  from  the 
yinited  value  of  the  other  two.  Thus,  XIV  represents  four- 
teen, XXIX  twenty-nine,  XCIV  ninety-four. 

TABLE   OF   ROMAN    NOTATION. 

1   is  One.  XVIII   is  Eighteen. 
II   "   Two.  XIX   -'   Nineteen. 

III  "   Three.  XX    "  Twenty. 

IV  "    Four.  XXI   "   Twenty-one. 
V   "    Five.  XXX   "   Thirty. 

VI  l<  Six.  XL   "   Forty. 

VII  "  Seven.  L   "   Fifty. 

VIII  "  Eight.  LX   "   Sixty. 

IX  "  Nine..  LXX    "    Seventy. 

X  "  Ten.  LXXX   "   Eighty. 

XI  "  Eleven.  XO   "   Ninety. 

XII  "  Twelve.  C   "   One  hundred. 

XIII  "  Thirteen.  CO   "  Two  hundred. 

XIV  "  Fourteen.  D   "   Five  hundred.    ^ 
XV  "  Fifteen.  DO    •'   Six  hundred.       - 

XVI  "   Sixteen.  M    "   One  thousand. 

XVII  "   Seventeen. 

Express  the  following  numbers  by  the  Roman  notation: 

1.  Fourteen.  6.  Fifty-one. 

2.  Nineteen.  7.  Eighty-eight. 

3.  Twenty-four.  8.  Seventy- three. 

4.  Thirty-nine.  9.  Ninety-five. 

5.  Forty-six.  10.  One  hundred  one. 


.  Employs  ten  characters  or  figures  to  express  numbers. 


10  SIMPLE  NUMBERS. 

Tim?, 

Figures,  01       234.56789 

Names   and  )     naught,  one,     two,  three,  four,  five,  six,  seven,  eight,  nine. 

values,  \  cip£r) 

30.  The  first  character  is  called  naught,  because  it  has 
no  value  of  its  own.  .  The  other  nine  characters  are  called 
significant  figures,  because  each  has  a  value  of  its  own. 

31.  As  we  have  no  single  character  to  represent  ten,  we 
express  it  by  writing  the  unit,  1,  at  the  left  of  the  cipher,  0, 
thus,  10.     In  the  same  manner  we  represent 

2  tens,        3  tens,    4  ten*,        5  tens    6  tens,         7  tens,  8  tens,       9  tens, 

or  or  ur  or  or  or  or  or 

twenty,      thirty,       forty,        fifty,      sixty,         seventy.          eighty,      ninety, 

20;       30;      40;      50;     60;        70;  80;       90. 

33.  When  a  number  is  expressed  by  two  figures,  the  right 
hand  figure  is  called  units,  and  the  left  hand  figure  tcn&. 
We  express  the  numbers  between  10  and  20,  thus : 

eleven,     twelve,  thirteen,  fourteen,   fifteen,  sixteen,  seventeen,  eighteen,  nineteen. 

11,      12,      13,       14,       15,     16,       17,       18,        19. 

In  like  manner  we  express  the  numbers  between  20  and 
30,  thus :    21,  22,  23,  24,  25,  26,  27,  28,  29,  &c. 
UThe  greatest  number  that  can  be  expressed  by  two  figures 
is  99. 

33.  We  express  one  hundred  by  writing  the  unit,  1,  at 
the  left  hand  of  two  ciphers  ;  thus,  100.  In  like  manner 
we  write  two  hundred,  three  hundred,  &c.,  to  nine  hundred. 
Thus: 

one          two         three         four          five  six        seven       eight       nine 

hundred,  hundred, hundred  , hundred,  hundred, hundred, hundrod,hundred, hundred. 

100,     200,     300,    400,     500,     600,    700,     800,     900. 

3J:.  When  a  number  is  expressed  by  three  figures,  the 
right  hand  figure  is  called  ujiits,  the  second  figure  tens,  and 
tho  left  hand  figure  Jmin/rc</s. 


NOTATION   AND   NUMERATION.  11 

As  the  ciphers  have,  of  themselves,  no  value,  but  are 
always  used  to  denote  the  absence  of  value  in  the  places  they 
occupy,  we  express  tens  and  units  with  hundreds,  by  writing, 
in  place  of  the  ciphers,  the  numbers  representing  the  tens 
and  units.  To  express  one  hundred  fifty,  we  write  1  hun- 
dred, 5  tens,  and  0  units ;  thus,  150.  To  express  seven 
hundred  ninety-two,  we  write  7  hundreds,  9  tens,  and  2 
units ;  thus, 


.§     3     § 

792 

The  greatest  number  that  can  be  expressed  by  thne 
figures  is  999. 

Express  the  following  numbers  by  figures  : — 

1.  Write  one  hundred  twenty-five. 

2.  Write  four  hundred  eighty-three. 

3.  Write  seven  hundred  sixteen. 

4.  Express  by  figures  nine  hundred.  W 

5.  Express  by  figures  two  hundred  ninety. 

6.  Write  eight  hundred  nine. 

7.  Write  five  hundred  five. 

8.  Write  five  hundred  fifty-seven. 

95.  We  express  one  thousand  by  writing  the  unit,  1,  at 
the  left  hand  of  three  ciphers ;  thus,  1000.  In  the  same 
manner  we  write  two  thousand,  three  thousand,  &c.,  to  nine 
thousand;  thus,  ^ 

one         two          three        four          five          six          seven       eight        nine 
thousand,  thousand,  thousand,  thousand,  thousand.thousand,  thousand,  thousand,  thousand. 

1000,  2000,  3000,  4000,  5000,  6000,  7000,  8000,  9000. 
9G.  When  a  number  is  expressed  by  four  figures,  the 
places,  commencing  at  the  right  hand,  are  units,  tens,  hun- 
dreds, thousands. 


12  SIMPLE    XUMBE11S. 

To  express  hundreds,  tens,  and  units  with  thousands,  we 
write  in  each  place  the  figure  indicating  the  number  we 
wish  to  express  in  that  place.  To  write  four  thousand  two 
hundred  sixty-nine,  we  write  4  in  the  place  of  thousands,  2 
in  the  place  of  hundreds,  6  in  the  place  of  tens,  9  in  the 
place  of  units ;  thus, 


4269 

The  greatest  number  that  can  be  expressed  by  four  figures 
is  9999. 

Express  the  following  numbers  by  figures : — 
^  1.  One  thousand  two  hundred. 

2.  Five  thousand  one  hundred  sixty. 

3.  Three  thousand  seven  hundred  forty-one. 

4.  Eight  thousand  fifty  six. 
«r    5.  Two  thousand  ninety. 

6.  Seven  thousand  nine. 

7.  One  thousand  one. 

8.  Nine  thousand  four  hundred  twenty-seven. 

9.  Four  thousand  thirty-five. 

10.  One  thousand  nine  hundred  four. 
Read  the  following  numbers : — 

11.  76;     128;     405;     910;     116;     3414;     1025. 
^12.  2100  ;  5047  ;  7009  ;  4670  ;  3997  ;     1001. 

27.  Next  to  thousands  come  tens  of  thousands,  and  next 
to  these  come  hundreds  of  thousands,  as  tens  and  hundreds 
come  in  their  order  after  units. 

Ten  thousand  is  expressed  by  removing  the  unit,  1,  oni 
place  to  the  left  of  the  place  of  thousands,  or  by  writing  it 


NOTATION    AND   NUMERATION.  13 

at  the  left  hand  of  four  ciphers  ;  thus,  10000  ;  and  one 
hundred  thousand  is  expressed  by  removing  the  unit,  1, 
still  one  place  further  to  the  left,  or  by  writing  it  at  the  left 
hand  of  five  ciphers  ;  thus,  100000.  We  can  express  thou- 
sands, tens  of  thousands,  and  hundreds  of  thousands  in  one 
number,  in  the  same  manner  as  we  express  units,  tens,  and 
hundreds  in  one  number.  To  express  five  hundred  twenty- 
one  thousand  eight  hundred  three,  we  write  five  in  the  sixth 
place,  counting  from  units,  2  in  the  fifth  place,  1  in  the 
fourth  place,  8  in  the  third  place,  0  in  the  second  place, 
(because  there  are  no  tens),  and  3  in  the  place  of  units  ; 
thus, 


*  §  tn 

II     *J    I    I     1    I 

5          21803 

The  greatest  number  that  can  be  expressed  by  Jive  figures 
is  99999  ;  and  by  six  figures,  999999.  •  9 

Write  the  following  numbers  in  figures  :  — 
%1.  Twenty  thousand. 
^2.  Forty-seven  thousand. 
^3.  Eighteen  thousand  one  hundred. 

Twelve  thousand  three  hundred  fifty. 

Thirty-nine  thousand  five  hundred  twenty-two. 
^6.  tS^een  thousand  two  hundred  six. 
Jk7.  Eleven  thousand  twenty-four. 
^8.  Forty  thousand  ten. 
^  9.  Sixty  thousand  six  hundred. 

Two  hundred  twenty  thousand. 

One  hundred  fifty-six  thousand. 

Eiglit  hundred  forty  thousand  three  hundred. 


14  SIMPLE   NUMBERS. 

Read  the  following  numbers: 

13.  5^06;       1^304;     96071;       &470 ;     20^410. 

14.  36.741;    40(\560;     13,061;     4^300;     lOOplO. 
•*-  For  convenience  in  reading  large  numbers,  we  may  point 
them  off,  by  commas,  into  periods  of  three  figures  each, 
counting  from  the  right  hand  or  unit  figure.     This  point- 
ing enables  us  to  read  the  hundreds,  tens,  and  units  in  each 
period  with  facility  as  seen  in  the  following 

NUMERATION    TABLE. 


2          £          2          £          |   ; 

•ggs    'S-S    •§•3    "§23    'gas 

Sis   5S§   5ls   5jJ    2S§ 

876,556,789,012,345 


fifth     fourth    third    second    first 
period,  period,  period,  period,  period. 

_  I.  ]ngures  occupying  different  places  in  a  number,  as 
units,  tens,  hundreds,  &c.,  are  said  to  express  different  or- 
ders of  units.  9 

29.  In  numerating,  or  expressing  numbers  verbally,^he 
various  orders  of  units  have  the  following  names  :  ^ 


ORDERS. 

NAMES. 

1st  order  is  called 

Units. 

2d    order  «       " 

Tens. 

3d   order  «       " 

Hundreds. 

4th  order  "       " 

Thousands. 

5th  order  "       « 

Tens  of  thousands. 

6th  order  "       " 

Hundreds  of  thousands. 

7th  order  "       " 

Millions. 

8th  order  "       " 

Tens  of  millions. 

9th  order  «       " 

Hundreds  of  millions. 

&c.,  &c. 

<&c.,  &c. 

NOTATION   AND   NUMERATION.  15 

"Write  and  read  the  following  numbers : — 

1.  One  unit  of  the  third  order,  two  of  the  second,  five  of 
the  first.  Ans.  125  ;  read,  one.  hundred  twenty-Jive. 

2.  Two  units  of  the  5th  order,  four  of  the  4th;  five  of  the 
2d,  six  of  the  1st. 

Ans.  24056;  read,  twenty-four  thousand  fifty-six. 

3.  Seven  units  of  the  4th  order,  five  of  the  third;  three 
of  the  2d,  eight  of  the  1st. 

4.  Two  units  of  the  7th  order,  nine  of  the  6th;  four  of 
the  3d,  one  of  the  1st,  seven  of  the  2d. 

5.  Three  units  of  the  6th  order,  four  of  the  2d. 

6.  Nine  units  of  the  8th  order,  six  of  the  7th,  three  of 
the  5th,  seven  of  the  4th,  nine  of  the  1st. 

7.  Four  units  of  the  10th  order,  six  of  the  8th,  four  of 
the  7th,  two  of  the  6th;  one  of  the  3d,  five  of  the  2d. 

8.  Eight  units  of  the  12th  order,  four  of  the  llth,  six  of 
the  10th,  nine  of  the  7th7  three  of  the  6th,  five  of  the  5th, 
two  of  the  3d,  eight  of  the  1st. 

30.  Since  the  number  expressed  by  any  figure  depends 
upon  the  place  it  occupies,  it  follows  that  figures  have  two 
values,  Simple  and  Local. 

31.  The  Simple  Value  of  a  figure  is  its  value  when  ta- 
ken alone ;  thus,  4,  7,  2. 

33.  The  Local  Value  of  a  figure  is  its  value  when  used 
with  another  figure  or  figures  in  the  same  number.  Thus, 
in  325,  the  local  value  of  the  3  is  300,  of  the  2  is  20,  and 
of  the  5  is  5  units. 

NOTE.— When  a  figure  occupies  units'  place,  its  simple  and  local  values 
are  the  same. 

33.  The  leading  principles  upon  which  the  Arabic  nota- 
tion is  founded  are  embraced  in  the  following 


16  SIMPLE   NUMBERS. 

GENERAL   LAWS. 

I.  AU  numbers  are  expressed  ~by  applying  the  ten  figure* 
to  the  different  orders  of  units. 

II.  The  different  orders  of  units  increase  from  right  to 
left,  and  decrease  from  left  to  right,  in  a  tenfold  ratio. 

III.  Evert/  removal  of  a  figure  one  place  to  the  left,  in- 
creases its  local  value  tenfold ;  and  every  removal  of  a  fig- 
ure one  place  to  the  right,  diminishes  its  local  value  tenfold. 

From  this  analysis  of  the  principles  of  Notation  and 
Numeration,  we  derive  the  following  rules : — 

RULE   FOR   NOTATION. 

I.  Beginning  at  the  left  hand,  write  the  figures  belonging 
to  the  highest  period. 

II.  Write  the  hundreds,  tens,  and  units,  of  each  success- 
ive period  in  their  order,  placing  a  cipher  wherever  an  order 
of  units  'is  omitted. . 

RULE   FOR   NUMERATION. 

I.  Separate  the  number  into  periods  of  three  figures  each, 
commencing  at  the  right  hand. 

II.  Beginning  at  the  left  hand,  read  each  period  sepa- 
rately, and  give  the  name  to  each  period,  except  the  last,  or 
period  of  units. 

34.  Until  the  pupil  can  write  numbers  readily,  it  may 
be  well  for  him  to  write  several  periods  of  ciphers,  point 
them  off,  over  each  period  write  its  name,  thus. 

Trillions,      Billions,      Millions,      Thousands,       Units. 

000,    000,    000,      000,      000 


NOTATION   AND   NUMERATION.  17 

and  then  write  the  given  numbers  underneath,  in.  their  ap- 
propriate places. 

EXERCISES   IN    NOTATION   AND    NUMERATION. 

*  4 

Express  the  following  numbers  by  figures :  — 

1.  Four  hundred  thirty-six. 

2.  Seven  thousand  one  hundred  sixty-four. 

3.  Twenty-six  thousand  twenty-six. 

4.  Fourteen  thousand  two  hundred  eighty. 

5.  One  hundred  seventy-six  thousand. 

6.  Four  hundred  fifty  thousand  thirty-nine. 

7.  Ninety-five  million. 

•8.  Four  hundred  •  eighty-three  million    eight  hundred 
sixteen  thousand  one  hundred  forty-nine. 

9.  Nine  hundred  thousand  ninety. 
10.  Ten  million  ten  thousand  ten  hundred  ten. 
Point  off,  numerate,  and  read  the  'following  numbers  :  — 


11.   8240. 
12.  400900. 
13.    308. 
14.  60720. 

%15.   111111. 
16.  57468139. 
17.    5628. 
18.  11111111. 

19.  .   370005. 
20.  9400706342. 
21.    38429526. 
22.  11111111111. 

23.  Write  seven  million  thirty-six. 

24.  Write  five  hundred  sixty-three  thousand  four. 

25.  Write  one  million  ninety-six  thousand. 

26.  A  certain  number  contains .  3  units  of  the  seventh 
order,  6  of  the  fifth,  4  of  the  fourth,  1  of  the  third,  5  of  the 
second,  and  2  of  the  first ;  what  is  the  number  ? 

27.  What  orders  of  units  are  contained  in  the  number 
290648  ? 


18 


SIMPLE    NUMBERS. 


ADDITION. 

35.  Addition  is  the  process  of  uniting  several  numbers 
of  the  same  kind  into  one  equivalent  number. 

SO.  TJie  Sum,  or  Amount,  is  the  result  obtained. 

ADDITION   TABLE. 


2  and    1  are    3 

3  and   1  are    4 

4  and    1  are    5 

5  and    1  are    6 

2  and    2  are    4 

3  and    2  are    5 

4  and    2  are    6 

5  and    2  are    7 

2  and   3  are    5 

3  and   3  are    6 

4  and    3  are    7 

5  and    3  are    8 

2  and    4  are    6 

Sand   4  are    7 

4  and    4  are    8 

5  and    4  are   9 

2  and    5  are    7 

3  and    5  are    8 

4  and    5  are    9 

5  and    5  are  10 

2  and    6  are    8 

3  and    6  are    9 

4  and   6  are  10 

5  and    6  are  11 

2  and    7  are    9 

Sand    7  are  10 

4  and    7  are  11 

5  and    7  are  12 

2  and    8  are  10 

Sand    8  are  11 

4  and    8  are  12 

5  and    8  are  13 

2  and    9  are  11 

3  and    9  are  12 

4  and    9  are  13 

5  and    9  are  14 

2  and  10  are  12 

3  and  10  are  13 

4  and  10  are  14 

5  and  10  are  15 

2  and  11  are  13 

3  and  11  are  14    j 

4  and  11  are  15 

5  aud  11  are  1  6 

2  and  12  are  14 

3  and  12  are  15   ' 

,  4  and  f  2  are  16 

5  and  12  ar^l" 

6  and    1  are    7 

17  and-^1  are    8 

8  and  .1  are    9 

;9£nd    1  are  10 

6  and   2  are    8 

7  and    2*|re    9 

8  and    2  are  10  "" 

9  and    2  arc  11 

6  and   3  are    9 

7  and    3  are  10  • 

.  8  and   3  are  11 

9  and    3  are  12 

6  and   4  are  10 

7  and    4  are  11 

'i8,a,nd    4  are  12 

•9  and    4  are  13 

6  and   6  are  11 

7  and    5  are  12 

Sand    5  are  13 

9  and    5  are  14  ' 

6  and    6  are  12 

7  and    6  are  13 

8  and    6  are  14 

9  and    6  are  15 

G  and    7  are  13 

7  and    7  are  14 

Sand    7*el5; 

9  and    7  are  16 

6  and    8  are  14 

7  and    8  are  15 

8  and    8  are  16 

9  and    8  are  17 

6  and    9  are  15 

7  and    9  are  16 

Sand   9  are  17 

9  and    9  are  18 

6  and  10  are  16 

7  and  10  are  17 

8  and  10  are  18 

9  and  10  are  19 

6  and  11  are  17 

7  and  11  are  18 

8  and  11  are  19 

9  and  11  are  20 

6  anil  12  are  18 

7  and  12  are  19 

8  and  12  are  20 

9  and  12  are  21 

10  and    1  are  11 

Hand    1  are  12 

12  and    1  are  13 

13  and    1  are  14 

10  and   2  are  12 

11  and    2  are  13 

12  and    2  are  14 

13  and    2  are  15 

10  and    3  are  13 

Hand    3  are  14 

12  and    3  are  15 

13  and    3  are  16 

10  and   4  are  14 

Hand    4  are  15 

12  aud   4  are    6 

13  aud    4  are  17 

10  and    5  are  15 

11  and    5  are  16 

12  and   5  are  17 

13  and    5  are  18 

10  and    G  are  16 

Hand    6  are  17 

12  and    6  are  18 

13  and    6  are  19 

10  and   7  arc  17 

11  and    7  are  18 

12  and    7  are  19 

13  and    7  are  20 

10  and    8  are  18 

11  and    8  are  19 

12  and    8  are  20 

13  and    8  are  21 

Wand   9  are  19 

11  and   9  are  20 

12  and    9  are  21 

13  and    9  are  22 

10  and  10  are  20 

11  and  10  are  21 

12  and  10  are  22 

13  and  10  are  23 

10  an'l  11  are  21 

11  "and  1  1  are  22 

12  and  11  are  23 

13  and  11  are  24 

10  and  12  are  22 

11  and  12  are  23 

12  and  12  are  24 

13  and  12  are  25 

ADDITION.  19 

MENTAL   EXERCISES. 

1.  A  farmer  paid  6  dollars  for  a  straw-cutter,  and  9  dol- 
lars for  a  plow ;  how  much  did  he  pay  for  both  ? 

ANALYSIS.  He  paid  the  sum  of  6  dollars  and  9  dollars,  which 
:s  15  dollars.  Therefore,  he  paid  15  dollars  for  both. 

2.  John  gave  4  apples  to  James,  8  to  Henry,  and  9  to 
Asa ;  how  many  did  he  give  to  all  ? 

8.  Gave  7  dollars  for  a  barrel  of  flour,  9  dollars  for  a 
hundred  weight  of  sugar,  and  6  dollars  for  a  tub  of  butter ; 
how  much  did  I  give  for  the  whole  ? 

4.  I  have  two  pear  trees ;  one  tree  produced  12  bushels 
of  pears,  aud  the  other  11  bushels ;  how  many  bushels  did 
both  produce  ? 

5.  A  man  bought  4  cords  of  wo^d  for  12  dollars,  and  7 
bushels  of  corn  for  5  dollars ;   how  much  did  he  pay  for 
both? 

6.  James  gave  11  cents  for  a  slate,  and  had  8  cents  left ; 
how  many  cents  had  he  at  first  ? 

7.  A  lady  paid  5  dollars  for  a  bonnet,  10  dollars  for  a 
shawl,  and  had  7  dollars  left ;  how  much  money  had  she  at 
first  ? 

8.  In  a  shop  are  8  men,  9  boys,  and  6  girls,  at  work ; 
how  many  persons  are  at  work  in  the  shop  ? 

9.  Rollin  bought  a  quire  of  paper  for  12  cents,  a  slate 
for  13  cents,  and  gave  10  cents  to  a  beggar  j   how  much 
money  did  he  pay  out  in  all  ? 

10.  A  man  bought  4  bushels  of  wheat  for  7  dollars,  18 
bushels  of  corn  for  11  dollars,  and  2  cords  of  wood  for  5 
dollars  ;  how  much  did  he  pay  for  the  whole  ? 

11.  A  farmer  has  6  cows  in  one  yard,  9  in  another,  and 
as  many   in  the   third  yard  as  in  both  the  others ;    how 
many  cows  has  he  ? 


20 


SIMPLE   NUMBERS. 


PROMISCUOUS   A 

2  and  5  are  how  many  ? 
6  and  2  are  how  many  ? 
2  and  4  are  how  many  ? 
8  and  9  are  how  many  ? 
9  and  4  are  how  many  ? 
4  and  7  are  how  many  ? 
8  and  6  are  how  many  ? 
6  and  8  are  how  many  ? 
7  and  2  are  how  many  ? 

DDITION   TABLE. 

7  and  9  are  how  many  ? 
6  and  5  are  how  many  ? 
3  and  6  are  how  many  ? 
4  and  4  are  how  many  ? 
7  and  8  are  how  many  ? 
9  and  3  are  how  many  ? 
5  and  4  are  how  many  ? 
3  and  8  are  how  many  ? 
5  and  6  are  how  many  ? 

3  and  9  are  how  many  ? 
4  and  5  are  how  many  ? 
9  and  8  are  how  many  ? 
8  and  5  are  how  many  ? 
4  and  9  are  how  many  ? 
5  and  4  are  how  many  ? 
2  and  7  are  how  many  ? 
7  and  5  are  how  many  ? 
5  and  2  are  how  many  ? 

5  and  8  are  how  many  ? 
3  and  7  are  how  many  ? 
6  and  4  are  how  many  ? 
7  and  6  are  how  many  ? 
6  and  8  are  how  many  ? 
9  and  5  are  how  many  ? 
8  and  3  are  how  many  ? 
9  and  6  are  how  many  ? 
5  and  7  are  how  many  ? 

6  and  9  are  how  many  ? 

7  and  7  are  how  many  ? 

8  and  4  are  how  many  ? 

8  and  7  are  how  many  ? 

4  and  8  are  how  many  ? 

9  and  2  are  how  many  ? 

5  and  3  are  how  many  ? 

6  and  6  are  how  many  ? 

7  and  4  are  how  many  ? 


4  and  6  are  how  many  ? 

7  and  3  are  how  many  ? 
2  and  8  are  how  many  ? 

5  and  9  are  how  many  ? 

8  and  8  are  how  many  ? 

6  and  7  are  how  many  ? 
5  and  5  are  how  many  ? 

9  and  7  are  how  many  *? 
9  and  9  are  how  many  1 


37.  The  Sign  of  Addition  is  the  perpendicular  cross,  -f, 
called  plus.     It  indicates  that  the  numbers  connected  by  it 
are  to  be  added ;  as  3  -f-  5  -j-  7,  read  3  plus  5  plus  7. 

38.  The  Sign  of  Equality  is  two  short,  parallel,  hori- 
zontal lines,  =.     It  indicates  that  the  'numbers,  or  combi- 
nation of  numbers,  connected  by  it  are  equal  j  as  4  -J-  8  = 
9  -f  3;  read  the  sum  of  4  plus  8  is  equal  to  the  sum  of  9 
plus  3 


ADDITION.  21 

CASE   I. 

39.  "When  the  amount  of  each  column  is  less 
than  10. 

I.  A  drover  bought  three  flocks  of   sheep.      The  first 
contained  232,  the  second  422,  and  the  third  245;  how 
many  did  he  buy  in  all  ? 

OPERATION.  ANALYSIS.  We  arrange  the  numbers  so 
«  .  j  that  units  of  like  order  shall  stand  in  the 
.gj'l  same  column.  We  then  add  the  columns 
232  separately,  for  convenience  commencing  at 

422  the  right  hand,  and  write  each  result  under 

245          *he  c°lun:)n  added.     Thus,  we  have  5  and  2 

and  2  are  9,  the  sum  of  the  units  ;  4  and  2 

Amount,  899  and  3  are  9,  the  sum  of  the  tens ;  2  and  4 

and  2  are  8,  the  sum  of  the  hundreds. — 
Hence,  the  entire  amount  is  8  hundreds  9  tens  and  9  units,  or 
899,  the  Answer. 

EXAMPLES  FOR  PRACTICE. 

(2.)  (3.)  (4.)  (5.) 

403  164  510;  234 

271  321  176  324 

124  510  203  140 

Ans.  798 

(6.)  (7.)  (8.)  (9.) 

1234  2041  3102  4100 

2405  3216  2253  1523 

5140  1500  4014  2041 

Ans.  8779 
10.  What  is  the  sum  of  421,  305  and  5162  ? 

II.  What  is  the  sum  of  3121,  436  and  2002  ? 


22  SIMPLE   NUMBERS. 

CASE  II. 

4O.  "When  the  amount  of  any  column  equals  or 
exceeds  10. 

1.  A  merchant  pays  397  dollars  for  freights,  476  dollars 
for  a  clerk,  and  873  for  rent  of  a  store  ;  what  is  the  amount 
of  his  expenses  ? 

OPERATION.  ANALYSIS.     We  arrange   the   numbers   so 

that  units  of  like  order  shall  stand  in  the  same 


.  „„  column.     We  then  add  the  first,  or  right  hand 

column,  and  find  the  sum  to  be*16  units,  or  1 

_  ten  and  6  units  ;  writing  the  6  units  under 

1746  the  column  of  units,  we  add  the  1  ten  to  the 

column  of  tens,  and  find  the  sum  to  be  24 

tens,  or  2  hundreds  and  4  tens;    writing  the  4  tens  under  tUe 

column  of  tens,  we  add  the  2  hundreds  to  the  column  of  hundreds, 

and  find  the  sum  to  be  17  hundreds,  or  1  thousand  and  7  hun- 

dreds ;  writing  the  7  hundreds  under  the  column  of  hundreds, 

and  the  1  in  thousands'  place,  we  have  the  entire  sum,  1746. 

NOTES.  —  1.  In  adding,  learn  to  pronounce  the  partial  results  without  naming  the 
figures  separately.  Thus,  in  the  operation  given  for  illustration,  say  3,  9,  16  ;  8, 
15,  24  ;  10,  14,  17. 

2.  —  When  1  he  sum  of  any  column  is  greater  than  9,  the  process  of  adding  the 
tens  to  the  next  column  is  called  carrying. 

41.  From  the  proceeding  examples  and  illustrations  we 
deduce  the  following 

RULE.  I.  Write  the  numbers  to  be  added  so  tliat  all  the 
wiits  of  the  same  order  shall  stand  in  the  same  column  ; 
that  is,  units  under  units,  tens  under  tens,  &c. 

II.  Commencing  at  units,  add  each  column  separately, 
and  write  the  sum  underneath,  if  it  be  less  than  ten. 

III.  If  the  sum  of  any  column  be  ten  or  more  than  ten, 
write  the  unit  figure  only,  and  add  tlie  ten  or  tens  to  the 
next  column, 

IV.  Write  the  entire  sum  of  the  last  column. 


•     ADDITION.  23 

PROOF.  .Begin  with  the  right  hand  or  unit  column,  and 
add  the  figures  in  each  column  in  an  opposite  direction  from 
that  in  which  they  were  first  added ;  if  the  two  results  agree, 
the  work  is  supposed  to  be  right. 

EXAMPLES   FOR   PRACTICE. 

(1.)  (2.)  (3.)  (4.)  (5.) 

inches,  feet.  pounds.  yards.  miles. 

142  325  75  407  1270 
325  46  276  96  342 
476  674  508  2584  79 

943     1045      859  3087  1691 

(6.)  (7.)  (8.)  (9.)  (10.) 

842  376  426  713  4761 

396  407  397  86  374 

472  862  450  345  83 

205      94  294  60  19 

11.  What  is  the  sum  of  912  -f  342  -f  187  -j-  46  ? 

Ans.  1487. 

12.  What  is  the  sum  of  214  -f  425  +90  -f  37  ? 

Ans.  766. 

13.  What  is  the  sum  of  56  feet,  450  feet,  and  680  feet  ? 

Ans.  1186  feet. 

14.  What  is  the  sum  of  1942  dollars,  and  685  dollars  ? 

15.  A  man  paid  375  dollars  for  a  span  of  horses,  160 
dollars  for  a  carriage,  and  87  dollars  for  a  harness ;    how 
much  did  he  pay  for  all  ?  Ans.  622  dollars. 

16.  A  man  traveled  476  miles  by  railroad,  390  miles  by 
steamboat,  and  120  miles  by  stage ;  how  many  miles  in  all, 
did  he  travel  ?  Ans.  986. 

17.  A  carpenter  built  a  house  for  2464  dollars,  a  barn  for 
496  dollars,  and  outhouses  for  309  dollars ;  how  much  did 
he  receive  for  building  all  ? 


SIMPLE    NUMBEKS. 


18.  A  merchant  bought  at  public  auction  520  yards  of 
broadcloth,  386  yards  of  muslin,  92  yards  of  flannel,  and 
156  yards  of  silk  ;  how  many  yards  in  all  ? 

19.  A  father  divided  his  estate  among  his  four  sons,  giv- 
ing each  2087  dollars  ;  what  was  the  amount  of  his  estate  ? 

20.  Three  persons  deposited  money  in  a  bank ;  the  first 
4780  dollars,  the  second    3042  dollars,  and  the  third  407 
dollars ;  how  much  did  they  all  deposit  ? 

21.  Five  men  engage  in  business  as  partners,  and  each 
puts  in  2375  dollars ;  what  is  the  whole  amount  of  capital 
invested  ?  ^        Ans.  11875  dollars. 


(22.)       (23.) 

765  347 

381  192 

976  763 

315  410 

169  507 

Am.  2606 

(26.)  (27.) 

767346  374205 

432761  108497 

386109  643024 

508763  879638 

Am.  2094979 
29.  3720  -f  647 


(24.) 

630 

815 

456 

307 

960 


(25.) 

4603 

7106 

972 

385 

64 


(28.) 
4076315 
5632870 
8219634 
3827692 


190  -f  82  =  how  many  ? 

Ans.  4639. 

30.  962  -|-  2161  -f  500  +  75  =  how  many  ? 

Ans.  3698, 

31.  4170  -j-  1009  -f  642  +  120  -f  18  =  how  many  ? 

32.  3000  +  47602  +  805  +  1266  +  76  =  how  many  ? 

33.  69  +  4030  -f  349  -f  1384  +  72  +  400  =  how  many? 


'ADDITION.  20 

34.  What  is  the  «um  of  two  thousand  eight  hundred  fif- 
ty-six, twelve  thousand  eighty-four,  seven  hundred  forty- 
two,  and  sixty-nine  ?  Arts.  14751. 

35.  What  is  the  amount  of  twenty  thousand  five  hundred 
ten,  six  thousand  nine  hundred  forty-four,  and  three  thou- 
sand two  hundred  ?  Ans.  30654. 

36.  What  is  the  sum  of  forty-seven  thousand  fifty,  nine 
thousand  one  hundred  six,  fourteen  hundred  ninety-two,  and 
five  hundred  twelve  ?  Ans.  58160. 

37.  What  is  the  sum  of  one  hundred  forty  thousand  three 
hundred  thirty-four,  seventy  nine-thousand  six  hundred  five, 
twenty-five  hundred  twenty-five,  and  three  thousand  sixty- 
nine?  Ans.  225533. 

38.  What  is  the  amount  of  five  hundred  thousand  five 
hundred  five,  eighty-four   thousand  two   hundred,  fifteen 
thousand  six  hundred  twenty,  and  seventeen  hundred  sev- 
enteen?   '  Ans.  602042. 

89.  How  many  men  in  an  army   consisting  of  26840 
infantry,  6370  cavalry,  3250  dragoons,  750   artillery,  and 
\  320  miners  1  Ans.  37530. 

«M:0.  A  merchant  deposited  125  dollars  in  bank  on  Mon- 
day, 91  on  Tuesday,  164  on  Wednesday^  200  on  Thursday, 
196  on  Friday,  and  73  on  Saturday;  how  much  did  he  de- 
posit during  the  week  1 

*41.  By  selling  a  farm  for  8586  dollars,  684  dollars  were 
lost ;  how  much  did  the  farm  cost  ? 

-*42.  If  I  were  born  in  1840,  when  will  I  be  63  years  old? 
.  43..  A  man  willed  his  estate  to  his  wife,  two  sons  and 
three  daughters;  to  his  daughters  he  gave  1565  dollars 
apiece,  to  his  sons  3560  dollars  each,  and  to  his  wife  4720 
*  dollars  ; .  how  much  was  his  estate  1  Ans.  16535  dollars. 

44.  A  man  engaging  in  trade,  gained  450  dollars  the  first 
2 


26  SIMPLE   NUMBERS. 

year,  684  dollars  the  second,  and  as  much  the  third  as  he 
gained  during  the  first  and  second  ;  how  much  was  his  whole 
gain  1  Ans.  2268  dollar?. 

45.  I   bought  three  village  lots  for  12570  dollars,  and 
sold  them  so  as  to  gain  745  dollars  on  each  lot ;  for  how 
much  did  I  sell  them  1  Avis.  14805  dollars. 

46.  A  has  3240   dollars,  B  has  5672  dollars,  and  C  has 
1000  more  than  A  and  B  together ;  how  many  dollars  have 
all  1  Ans.  18824  dollars. 

47.  A  man  was  32  years  old  when  his  son  was  born ;  how 
old  will  he  be  when  his  son  is  3G  years  old  1     Ans.  68  years. 

48.  The  Old  Testament  contains  39  books,  929  chapters, 
23214  verses,  592439  words,  and  2728100  letters;  the  New 
Testament  contains  37  books,  269  chapters,  7959  verses, 
181153  words,  and  838380  letters;  what  is  the  total  num- 
ber of  each  in  the  Bible  1  ^ 

Ans.  76  books,   1198  chapters,  31173  verses,  773592 
words,  and  3566480  letters. 

49.  The  number  of  immigrants  landed  in  New  York  in 
1858  was  78589, in  1859,  79322,  and  in  1860,  103621 ; 
what  was  the  total  number  landed  in  the  three  years  ? 

Ans.  261532. 

50.  In  1860,  the  population  of  New  York  was  814277, 
of  Philadelphia  568034,  of  Boston  177902,  of  New  Orleans 
170766,  of  St.  Louis  162179,  of  Chicago  109429,  and  3t 
Cincinnati  160000 ;  what  was  the  total  population  of  these 
cities  1  Ans.  2162587  ^ 

51.  In  the  year  1856,  the  United  States  exported  molasses 
to  the  value  of  154630  dollars;  in   1857,108003   dollars; 
in  1858,  115893  dollars;  what  was  the  value  of  the  molas- 
ses exported  in  those  three  years'?      Ans.  378526  dollar*. 

52.  During  the  same  years,  respectively,  the  United  States 


ADDITION.  27 

exported  tobacco  to  the  value  of  1829207  dollars,  1458553 
dollars,  and  2410224  dollars ;  what  was  the  total  value  of  the 
tobacco  exported  in  those  years  1  Ans.  5697984  dollars. 

53.  How  many  miles  from  the  southern  extremity  of  Lake 
Michigan  to  the  Gulf  of  St.  Lawrence,  .passing  through 
Lake  Michigan,  330  miles ;  Lake  Huron,  260  miles ;  River 
St.  Clair,    24   miles;  Lake   St.  Clair,    20    miles;    Detroit 
River,  23  miles;  Lake  Erie,  260  miles;  Niagara  River,  34 
miles ;  Lake  Ontario,  180  miles ;  and  the  River  St.  Law- 
rence, 750  miles?  Ans.  1881  miles. 

54.  At  the  commencement  of  the  year  1858  there  were 
in  operation  in  the  New  England  States,  3751  miles  of  rail- 
road ;   in  New  York,  2590  miles ;  in   Pennsylvania,  2546 ; 
in  Ohio,  2946;  in  Virginia,  1233  ;  in  Illinois,  2678  ;  and 
in  Georgia,  1233 ;  what  was  the  aggregate  number  of  miles 
in  operation  in  all  these  States  1  Ans.  16977. 

55.  The  number  of  pieces  of  silver  coin  made  at  the  Uni- 
ted States  Mint  at  Philadelphia  in  the  year  1858,  were  as 
follows:  4628000  half  dollars,  10600000  quarter  dollars, 
690000  dimes,  4000000  half  dimes,  and  1266000  three- 
cent  pieces ;  what  was  the  total  number  of  pieces  coined  1 

Ans.  21184000. 

(56.)  (57.)  (58.)  (59.) 

344  843  1186  81988 

388  738  513  380167 

613  237  740.  108424 

803  218  1820  193686 

825  347  955  144225 

412  288  736  112558 

322  483  810  107481 

886  753  511  176826 

620  834  1179  145851 

5213  8450     1451206 


28 


SIMPLE    NUMBERS. 


(60.) 

35938 
49172 
56546 
82564 
69789 
47321 
77563 
83563 
54973 
38137 
54246 
95864 
48135 
37975 
48467 


(61.) 

47197 
63956 
85678 
35495 
16457 
94667 
76463 
34698 
17179 
93965 
81367 
29787 
79826 
31275 
59689 


(62.) 

12380 
98795 
23442 
87639 
91758 
19347 
81731 
29342 
75659 
35446 
98237 
12845 
87677 
23444 
39878 


(63.) 

456568 
754712 
567346 
543678 
3427G6 
768345 
563875 
547427 
945956 
165675 
756431 
354747 
543864 
567456 
621367 


(64.) 
768856 
674387 
978874 
567678 
568594 
639678 
669657 
594886 
695756 
789568 
689689 
638786 
675968 
958789 
769896 
153674 
331767 
355989 


(65.) 
576654 
678456 
754543 
786567 
964432 
699678 
978321 
678789 
564673 
895437 
569128 
678982 
869771 
668339 
956234 
195876 
957412 
573375 


(66.) 
987654 
123456 
876864 
234246 
765183 
345927 
654678 
456432 
345719 
765391 
673123 
437987 
566789 
544321 
891389 
219721 
625247 
431321 


(67.) 
9873785 
1239564 
7591074 
3517569 
8598674 
2513756 
8454210 
7656754 
5467856 
5645781 
7893344 
8216677 
4569911 
6543344 
9576677 
1539900 
6662233 
4235566 


11522492 


13046667 


9945448 


99796675 


SUBTRACTION. 


29 


SUBTRACTION. 

42.  Subtraction  is  the  process  of  determining  the 
difference,  between  two  numbers  of  the  same  unit  value. 
43.  The  Difference  or  Remainder  is  the  result  obtained. 

SUBTRACTION   TABLE. 


1  from    2  leaves    1 

2  from    3  leaves    1 

3  from    4  leaves    1 

4  from    5  leaves    1 

1  from    3  leaves    2 

2  from    4  leaves    2 

3  from    5  leaves    2 

4  from    6  leaves    2 

1  from    4  leaves    3 

2  from    5  leaves    3 

3  from    6  leaves    3 

4  from    7  leaves    3 

1  from    5  leaves    4 

2  from    6  leaves    4 

3  from    7  leaves    4 

4  from    8  leaves    4 

1  from    6  leaves    5 

2  from    7  leaves    6 

3  from    8  leaves    5 

4  from    9  leaves    5 

1  from    7  leaves    6 

2  from    8  leaves    6 

3  from    9  leaves    6 

4  from  10  leaves    6 

1  from    8  leaves    7 

2  from    9  leaves    7 

3  from  10  leaves    7 

4  from  11  leaves    7 

1  from    9  leaves    8 

2  from  10  leaves    8 

3  from  11  leaves    8 

4  from  12  leaves    8 

1  from  10  leaves    9 

2  from  11  leaves    9 

3  from  12  leaves    9 

4  from  13  leaves    9 

1  from  11  leaves  10 

2  from  12  leaves  10 

3  from  13  leaves  10 

4  from  14  leaves  10 

5  from    6  leaves    1 

6  from    7  leaves    1 

7  from    8  leaves    1 

8  from    9  leaves    1 

5  from    7  leaves    2 

6  from    8  leaves    2 

7  from    9  leaves    2 

8  from  10  leaves    2 

5  from    8  leaves    3 

6  from    9  leaves    3 

7  from  10  leaves    3 

8  from  11  leaves    3 

5  from    9  leaves    4 

6  from  10  leaves    4 

7  from  11  leaves    4 

8  from  12  levves    4 

5  from  10  leaves    5 

6  from  11  leaves    5 

7  fr»m  12  leaves    5 

8  from  13  leaves    5 

5  from  11  leaves    6 

6  from  12  leaves    6 

7  from  13  leaves   6 

8  from  14  leaves    6 

5  from  12  leaves    7 

6  from  13  leaves    7 

7  from  14  leaves    7 

8  from  15  leaves    7 

5  from  13  leaves    8 

6  from  14  leaves    8 

7  from  15  leaves    8 

8  from  16  leaves    8 

5  from  14  leaves    9 

6  from  15  leaves    9 

7  from  16  leaves    9 

8  from  17  leaves    9 

5  from  15  leaves  10 

6  from  16  leaves  10 

7  from  17  leaves  10 

8  from  18  leaves  10 

9  from  10  leaves    1 

10  from  11  leaves  1 

11  from  12  leaves   1 

12  from  13  leaves    1 

9  from  11  leaves    2 

10  from  12  leaves  2 

11  from  13  leaves   2 

12  from  14  leaves    2 

9  from  12  leaves    3 

10  from  13  leaves  3 

11  from  14  leaves   3 

12  from  15  leaves    3 

9  from  13  leaves    4 

10  from  14  leaves  4 

11  from  15  leaves   4 

12  from  16  leaves    4 

9  from  14  leaves    5 

10  from  15  leaves  5 

11  from  16  leaves   5 

12  from  17  leaves    6 

9  from  15  leaves    6 

10  from  1(3  leaves   6 

11  from  17  leaves   6 

12  from  18  leaves   6 

9  from  16  leaves    7 

10  from  17  leaves  7 

11  from  18  leaves   7 

12  from  19  leaves    7 

9  from  17  leaves    8 

10  from  18  leaves  8 

11  from  19  leaves  8 

12  from  20  leaves    8 

9  from  18  leaves    0 

10  from  19  leaves  9 

11  from  20  leaves   9 

12  from  21  leaves    9 

9  from  19  leaves  10 

10  from  20  leaves  10 

11  from  21  leaves  10 

12  from  22  leaves  10 

80  SIMPLE    NUMBERS. 

MENTAL    EXERCISES. 

1.  A  grocer  having  20  boxes  of  lemons,  sold  12  boxes  \ 
how  many  boxes  had  he  left] 

ANALYSIS.     He  had  left  the  difference  between  20  boxes  and 
12  boxes,  which  is  8  boxes.     Therefore,  he  had  8  boxes  left. 

2.  If  a  man  earn  12  dollars  a  week,  and  spend  7  for  pro- 
visions, how  many  dollars  has  he  left  f( 

3.  If  I  borrow  15  dollars,  and  pay  9  dollars,  how  many 
dollars  remain  unpaid  ? 

4.  John  had  11  marbles,  and  lost  5  of  them;  how  many 
had  he  left? 

5.  From  a  cistern  containing  22  barrels  of  water,  9  barrels 
leaked  out ;  how  many  barrels  remained  1 

6.  In  a  school  are  24  boys  and  Ii2  girls ;  hoW  many  more 
boys  than  girls  ? 

7.  From  a  piece  of  cloth  containing  17  yards,  8  yards 
were  cut;  how  many  yards  remained  ? 

8.  Grin   paid   15   dollars  for  a  coat,  and  9  dollars  for  a 
pair  of  pantaloons ;  how  niuch  more  did  he  pay  for  the  coat 
than  for  the  pantaloons  ?  „ ' 

9.  Cora  is  23  years  old,  and   her   brother  is   10  years 
younger ;   how  old  is  her  brother  ? 

10.  A  jeweler  bought  a  watch  for  11  dollars,  and  sold  it 
for  1 8  dollars ;  how  much  did  he  gain  ? 

11.  A  boy  gave  21   cents  for  some  pictures,  which  were 
worth  no  more  than  17  cents;   how  much  more  than  their 
value  did  he  give  for  them  1 

12.  A  grocer  bought  a  barrel  of  sugar  for  16   dollars, 
but  it  not  proving  as  good  as  he  expected,  he  sold  it  for  11 
dollars ;  how  much  did  he  lose  on  it  ? 


SUBTRACTION. 


PROMISCUOUS    SUBTRACTION    TABLE. 


5  from  1  i  how  many  ? 
5  from     9  how  many  ? 
9  from  10  how  many  ? 
6  from     7  how  many  ? 
7  from  12  how  many  ? 
9  from  12  how  many  ? 
5  from  10  how  many  ? 
6  from  11  how  many  ? 

6  from  14  how  many? 
8  from  15  how  many? 
5  from  11  how  many? 
7  from  10  how  many  ? 
3  from  13  how  many  ? 
9  from  11  how  many  ? 
6  from  12  how  many  ? 
8  from  10  how  many  ? 

8  from     9  how  many  ? 
7  from  16  how  many  ? 
2  from  11  how  many? 
5  from     8  how  many  ? 
9  from  14  how  many  ? 
9  from  13  how  many  ? 
7  from     9  how  many  ? 
2  from  10  how  many  ?    . 

4  from  11  how  many  ? 
3  from  10  how  many  ? 
5  from  12  how  many  ? 
7  from  13  how  many  ? 
8  from  12  how  many  ? 
9  from  16  how  many  ? 
6'  from  13  how  many  ? 
4  from  12  how  many? 

8  from  16  how  many  ? 

9  from  1 5  how  many  ? 
7  from  11  how  many? 

3  from  12  how  many  ? 
6  from  15  how  many  ? 
9  from  18  how  many  ? 
6  from  10  how  many  ? 

4  from  13  how  many  ? 


7  from  15  how  many  ? 

8  from  17  how  many  ? 

4  from  10  how  many  ? 

7  from  14  how  many  ? 
3  from  11  how  many  ? 

5  from  13  how  many  ? 

9  from  17  how  many  ? 

8  from  14  how  many? 

44.  The  Minuend  is  the  number  to  be  subtracted  from. 

45.  The  Subtrahend  is  the  number  to  be  subtracted. 

46.  The   Sign  of   Subtraction  is    a   short   horizontal 
line,  — ,  called  minus.     When  placed  between  two  numbers, 
it  denotes  that  the  one  after  it  is  to  be  taken  from  the 
one  before  it.     Thus,  8 — 6=2,  .is  read  8  minus  6  equals 
2,  and  denotes  that  6,  the  subtrahend,  taken  from  8,  the 
minuend)  equals  2,  the  remainder. 


32  SIMPLE    NUMBERS. 

CASE  i. 

47.  When  no  figure  in  the  subtrahend  is  greater 
than  the  corresponding  figure  in  the  minuend. 
1.  From  574  take  323. 

OPERATION.  ANALYSIS.    We  write  the  less  num 

Minuend,  574       her  under  the  greater,  with  units  un- 

Subtrahend,  823          d 


777       draw  a  line  underneath.     Then,  be- 
der'  ginning  at  the  right  hand,  we  sub- 

tract  separately  each  figure  of  the  subtrahend  from  the  figure 
above  it  in  the  minuend.  Thus,  3  from  4  leaves  1,  which  is  the 
difference  of  the  units;  2  from  7  leaves  5,  .the  difference  of  the 
tens  ;  3  from  5  leaves  2,  the  difference  of  the  hundreds.  Hence, 
we  have  for  the  whole  difference,  2  hundreds  5  tens  and  1  unit, 
or  251. 

EXAMPLES    FOR   PRACTICE. 

(1.)  (2.)  (3.)  (40 

Minuend,        876  349  637  508 

Subtrahend,      435  212  431  104 

Remainder,        441  137  206  404 

(5.)  (6.)  (7.)  (8.) 
987  753  438  695 
647  502  421  535 

340       251        17       160 

(9.)  (10.)  (11.)  (12.) 
From  7642  8730  2369  9786 
Take  3211  6430  2104  3126 

4431      2300       265      6660 


SUBTRACTION.  33 

Remainders. 

13.  From  4376  take  1254.  3122. 

14.  From  70342  take  5*0130.  20212. 

15.  From  137647  take  16215.  121432. 

16.  Subtract  32014  from  86325.  54311. 

17.  Subtract  217356  from  719568.  502212. 

18.  437615  —  213502  =  how  many  ?  224113.  , 

19.  732740  — 11520  =  how  many  ?  721220 

20.  2042674  —  32142  =  how  many  ? 

21.  8461203  —  7161003  =  how  many  ? 

22.  From  three  thousand  two  hundred  seventy-six,  take 
two  thousand  one  hundred  forty-three. 

23.  From  one  hundred  eighty-three  thousand  four  hun- 
dred sixty,  take  fifty-two  thousand  one  hundred  fifty. 

Am.  131310. 

24.  A  man  bought  a  piece  of  property  for  7634  dollars, 
and  sold  the  same  for  3132  dollars ;  how  much  did  he  lose  ? 

Ans.  4502  dollars. 

25.  A  merchant  sold  goods  to  the  amount  of  41763  dol- 
lars, and  by  so  doing  gained  11521  dollars  ;  how  much  did 
the  goods  cost  him  1  Ans.  30242  dollars. 

26.  A  drover  bought  3245  sheep,  and  sold  1249  of  them 
how  many  sheep  had  he  left  ? 

27.  A  general  before  commencing  a  battle  had  18765  men 
in  his  army ;  after  the  battle  he  had  only  8530 ;  how  many 
men  did  he  lose  1  Ans.  10235. 

28.  Two  persons  bought  a  block  of  buildings  for  69524 
dollars ;  one  paid  47321  dollars ;  how  much  did  the  other 
pay  1  Ans.  22203  dollars. 

29.  If  a  man's  annual  income  is  13460  dollars,  and  hig 
expenses  are  3340  dollars,  how  much  does  he  save  ? 

Ans.  10120  dollars. 


34  SIMPLE  NUMBEKS. 

CASE   II. 

48.  When  any  figure  in  the  subtrahend  is  greater 
than  the  corresponding  figure  in  the  minuend. 

I.  From  846  take  359. 

OPERATION.       ANALYSIS.     Since  we  cannot  take  9  units  from 
•|  ^  _jo          6  units,  we  add  10  units  to  6  units,  making  16 
J2  §  0          units;  and  9  units  from  16  units  leave  7  units. 
But  as  we  have  added  10  units,  or  1  ten  to  the 
minuend,  we  shall  have  a  remainder  1  ten  too 
'.  Q  I          large,  to  avoid  which,  we  add  1  ten  to  the  5  tens 
in  the  subtrahend,  making  6  tens.     We  can  not 
take  6  tens  from  4  tens;  so  we  add  10  tens  to  4,  making  14 
tens  ;  6  tens  from  14  tens  leave  8  tens.     Now,  having  added  10 
tens,  or  1  hundred,  to  the  minuend,  we  shall  have  a  remainder  1 
hundred  too  large,  unless  we  add  1  hundred  to  the  3  hundreds 
in  the  subtrahend,  making  4  hundreds ;  4  hundreds  from  8  hun- 
dreds leave  4  hundreds,  and  we  have  for  the  total  remainder,  487. 

NOTE.    The  process  of  adding  10  to  the  minuend  is  sometimes  called  borrowing 
10;  and  that  of  adding  1  to  the  next  figure  of  the  subtrahend,  carrying  one  . 

4tO.  From  the  preceding  example  and  illustration  we 
have  the  following  general 

RULE.  I.  Write  the  less  number  under  the  greater,  plac- 
ing units  of  the  same  order  in  the  same  column. 

II.  Beginning  at  the  right  hand,  take  each  figure  of  the 
subtrahend  from  the  figure  above  it,  and  write  the  result  un 
dern&ath. 

III.  If  any  figure  in  the  subtrahend  be  greater  than  the 
corresponding  figure  above  it,  add  10  to  that  upper  figure 
before  subtracting,  and  then  add  1  to  the  next  left  hand  fig- 
ure of  the  subtrahend. 

PROOF.  1st.  Ad'd  the  remainder  to  the  subtrahend;  the 
sum  will  be  equal  to  the  minuend.  Or, 

2d.  Subtract  the  remainder  from  the  minuend;  the  dif- 
ference will  be  equal  to  the  subtrahend. 


SUBTRACTION. 


36 


EXAMPLES  EOR  PRACTICE. 


Minuend, 

00 

753 

(20 
6731 

(3.) 
3248 

(4.) 
90361 

Subtrahend, 
Remainder, 

469 

2452 

1863 

6284 

284 

4279 

1385 

84077 

(5.) 

miles. 

(6.) 

bushels. 

(70 

dollars. 

(8.) 

feet. 

3146 

19472 

45268 

24760 

2529 

14681 

24873 

3478 

617 

4791 

20395 

23282 

(9.) 

rods. 

40307 

(10.) 

days. 

14605 

(11.) 

acres. 

23617 

(12.) 

gallons. 

980076 

38421 
1886 

8341 

14309 

94087 

6264 

9308 

885989 

(13.) 

men. 

17380 

(14.) 

sheep. 

282731 

(15.) 

barrels. 

80014 

(16.) 

tons. 

941000 

3417 

90756 

43190 

5007 

13963 

191975 

36824 

935993 

(17.) 
8077097 

(18.) 
3000001 

(19.) 

1970000 

1829164 

2199077 

1361111 

1247933 


800924 


608889 


36 


SIMPLE  NUMBERS. 


(20.) 

6000000 

999999 

500Q001 


(21.) 

8000800 

457776 

7543024 


(22.) 

103810040 
91300397 

12509643 
Ans.  224130. 


23.  234100  —  9970  ==  how  many  ? 

24.  3749001— 349623=how  many  ? 

25.  4000320  —  20142  =  how  many  ? 

26.  14601896  —  764059  =  how  many  ? 

27.  From  4716359  take  2740714.         Ans.  1975645. 

28.  From  7867564  take  2948675.         Ans.  4918889. 

29.  From  7788996  take  849842.  Ans.  6939154. 

30.  From  1073563  take  182000.  Ans.  891563. 

31.  From  1111111  take  111112.  Ans.  999999. 
82.  Subtract  1234509  from  8643587.     Ans.  7409078. 

33.  Subtract  1000  from  1100000.         Ans.  1099000. 

34.  Subtract  100701  from  846587. 

35.  Subtract  432986702100  from  539864298670. 

36.  Subtract  29176807982  from  86543298765. 

37.  A  speculator  boughf  wild  lands  for  10580  dollars,and 
sold  them  for  7642  dollars;  how  much  did  he  lose  ? 

Ans.  2938  dollars. 

38.  Napoleon  the  Great  was  born  in  1769,  and  died  in 
1821 ;  how  old  was  he  at  his  death  ?          Ans.  52  years. 

39.  Gunpowder  was  invented  in  1330;  and  printing  in 
1440  ;  how  many  years  between  the  two  ?         Ans.  110. 

40.  George  Washington  was  born  in  1732,  and  died  in 
1799  ;  how  old  was  he  at  his  death  ?          Ans.  67  years. 

41.  The  first  newspaper  published  in  America  was  issued 
at  Boston  in  1704  ;  how  long  Was  that  before  the  death  of 
Benjamin  Franklin,  which  occurred  in  1790  ? 

Ans.  86  years. 


PROMISCUOUS   EXAMPLES.  37 

42.  The  first  steamboat  in  the  United  States,  built  by 
Robert  Fulton,  in  1807,   made  a  trip    from   New  York 
to  Albany  in  33  hours ;  how  many  years  from  that  time  to 
the  visit  of  the  Great  Eastern  to  this  country  in  1860  ? 

Ans.  53  years. 

43.  Queen  Victoria  was  born  in  1819 ;  what  will  be  her 
age  in  1862  ?  Ans.  43  years. 

44.  The   United  States  contain   2983153  square  miles, 
and  the  British  North  American  Provinces  3125401  square 
miles.     How  many  square  miles  does  the  latter  country  ex- 
ceed the  former  ?  Ans.  142248. 

EXAMPLES    COMBINING   ADDITION   AND    SUBTRACTION. 

1.  A  farmer  having  450  sheep,  sold  124  at  one  time,  and 
96  at  another  ;  how  many  had  he  left  ?  Ans.  230. 

2.  If  a  man's  income  is  175  dollars  a  month,  and  he  pays 
25  dollars  for  rent,  44  dollars  for  provisions,  and  18  dollars 
for  other  expenses,  how  much  will  he  have  left  ? 

Ans.  88  dollars. 

3.  A  man  gave  his  note  for  3245  dollars.     He  paid  at 
one  time  780  dollars,  and  at  another  484  dollars ;  how  much 
remained  unpaid  ?  Ans.  1981  dollars? 

4.  A  man  paid  140  dollars  for  a  horse  and  165  dollars  for  a 
carriage.     He  afterward  sold  them  both  for  300  dollars  ]  did 
he  gain  or  lose,  and  how  much  ?         Ans.  Lost  5  dollars. 

5.  A  flour  merchant  having  700  barrels  of  flour  on  hand, 
sold  278  barrels  to  one  man,  and  142  to  another ;  how  many 
barrels  had  he  left  ?      ~~  Ans.  280  barrels. 

6.  Three  men  bought  a  farm  for  9840  dollars.     The  first 
paid  2672  dollars,  the  second  paid  3089  dollars,  and  the 
third  the  remainder ;  how  much  did  the  third  pay  ? 

Ans    4079  dollars. 


88  SIMPLE  NUMBERS. 

7.  A  man  bought  a  house  for  1500  dollars,  and  having 
expended  315  dollars  for  repairs,  sold  it  for  2000  dollars ; 
how  much  was  his  gain  ?  Ans.  185  dollars. 

8.  Henry  Jones  owns  property  to  the  amount  of  36748  dol- 
lars, of  which  he  has  invested  in  real  estate  12850  dollars,  in 
personal  property  9086  dollars,  and  the  remainder  he  has  in 
bank ;  how  much  has  he  in  bank  ?     Ans.  14812  dollars. 

9.  A  grocer  bought  275  pounds  of  butter  of  one  farmer, 
and  318  pounds  of  another;  he  afterward  sold  210  pounds 
to  one  customer,  and  97  to  another ;  how  many  pounds  had 
he  left  ?  Ans.  286  pounds. 

10.  A  man  deposited  in  bank  10476  dollars ;  he  drew 
out  at  one  time  .2356  dollars,  at  another  1242,  and  at  anoth- 
er 737  dollars ;  how  much  had  he  remaining  in  bank  ? 

Ans.  6141  dollars. 

11.  Borrowed  of  my  neighbor  at  one  time  680  dollars,  at 
another  time  910  dollars,  and  at  another  time  218  dollars. 
Having  paid  him  1309   dollars,  how  much  do  I  still  owe 
him  ?  Ans.  499  dollars. 

12.  A  man  bought  3  lots ;  for  the  first  he  paid  2480  dol- 
lars, for  the  second  3137  dollars,  and  for  the  third  as  much 
as  for  the  other  two ;  he  afterward  sold  them  for  15000 
dollars ;  how  much  was  his  gain  ?         Ans.  3766  dollars. 

13.  A  farmer  raised  1864  bushels  of  wheat,  and  1129 
bushels  of  corn.     Having  sold  1340  bushels  of  wheat,  and 
1000  bushels  of  corn,  how  many  bushels  of  each  has  he  re- 
mainiDg  ?  Ans.  524  bushels,  and  129  bushels. 

14.  A  gentleman  worth  25800  dollars,  bequeathed  his  es- 
tate so  that  each  of  his  two  sons  should  have  9400  dollars, 
and  his  daughter    the  remainder.     How  much   was  the 
daughter's  portion  ? 


MULTIPLICATION. 


39 


MULTIPLICATION. 

50.  Multiplication  is  the  process  of  taking  one  of  two 
given  numbers  as  many  times  as  there  are  un*its  in  the  other. 

51.  The  Product  is  the  result  obtained. 

MULTIPLICATION   TABLE. 


Once      1  is       1 

2  times    1  are    2 

3  times    1  are   3 

4  times  1  are    4  i 

Once     2  is       2 

2  times    2  are   4 

3  times    2  are   6 

4  times    2  are    8 

Once     3  is       3 

2  times    3  are    6 

8  times   8  are   9 

4  times    3  are  12 

Once     4  is       4 

2  times   4  are    8 

3  times   4  are  12 

4  times    4  are  16 

Once     5  is       6 

2  times    5  are  10 

3  times    5  are  15 

4  times    5  are  20 

Once     6  is       6 

2  times    6  are  12 

3  times    6  are  18 

4  times    6  are  24 

Once     7  ia       7 

2  times    7  are  14 

3  times    7  are  21 

4  times    7  are  28 

Once     8  is       8 

2  times    8  are  16 

3  times    8  are  24 

4  times    8  are  32 

Once     9  is       9 

2  times    9  are  18 

3  times    9  are  27 

4  times    9  are  36 

Once   10  ia     10 

2  times  10  are  20 

3  times  10  are  30 

4  times  10  are  40 

Once   11  is     11 

2  times  11  are  22 

3  times  11  are  33 

4  times  11  are  44 

.  Once    12  is     12 

2  times  12  are  24 

8  times  12  are  36 

4  times  12  are  48 

5  times    1  are    5 

6  times    1  are    6 

7  limes    1  are    7  . 

8  times    1  are   8 

5  times    2  are  10 

6  times    2  are  12 

7  times    2  are  14 

8  times    2  are  16 

5  times    3  are  15 

6  times    3  are  18 

7  times    3  are  21 

8  times    3  are  24 

5  times    4  are  20 

6  times    4  are  24 

7  times   4  are  28 

8  times    4  are  32 

5  times    5  are  25 

6  times   5  are  30 

7  times    6  are  35 

8  times    5  are  40 

5  times    6  are  30 

6  times    6  are  36 

7  times    6  are  42 

8  times    6  are  48 

5  times    7  are  35 

6  times    7  are  42 

7  times    7  are  49 

8  times    7  are  56 

5  times    8  are  40 

6  times   8  are  48 

7  times    8  are  56 

8  times    8  are  64 

5  times    9  are  45 

6  times    9  are  54 

7  times    9  are  63 

8  times    9  are  72 

5  times  10  are  50 

6  times  10  are  60 

7  times  10  are  70 

8  times  10  are  80 

5  times  11  are  55 

6  times  11  are  66 

7  times  11  are  77 

8  times  11  are  88 

5  times  12  are  60 

6  times  12  are  72 

7  times  12  are  84 

8  times  12  are  96 

9  times    1  are      9 

10  times    1  are    10 

11  times    1  are    11 

12  times    1  are    12 

9  times    2  are   18 

10  times    2  are    20 

11  times   2  are    22 

12  times    2  are    24 

9  times    3  are    27 

10  times    3  are    30 

11  times    3  are    33 

12  times   3  are   36 

9  times    4  are   36 

10  times    4  are    40 

lltimea    4  are    44 

12  times   4  are    48 

9  times   5  are    45 

10  times    5  are    50 

11  times    5  are    55 

12  times    5  are    66 

9  times    6  are    54 

10  times    6  are    60 

11  times    6  are    66 

12  times    6  are    72 

9  times    7  are    63 

10  times    7  are    70 

11  times    7  are    77 

12  times    7  are    84 

9  times    8  are    72 

10  times    8  are    80 

11  times    8  are    88 

12  times    8  ane    96 

9  times    9  are    81 

10  times  9  are    90 

11  times    9  are    99 

12  times    9  are  108 

9  times  10  are    90 

10  times  10  are  100 

11  times  10  are  110 

12  times  10  are  120 

9  times  11  are    99 

10  times  11  are  110 

11  times  11  are  121 

12  times  11  are  132 

9  times  12  are  10S 

10  times  12  are  120 

11  times  12  are  132 

12  times  12  are  144 

4:0  SIMrLE  NUMBERS. 

MENTAL   EXERCISES. 

1.  At  9  cents  a  pound,  what  will  7  pounds  of  sugar  cost/ 

ANALYSIS.  Since  one  pound  costs  9  cents,  7  pounds  will  cost 
7  times  9  cents,  or  63  cents.  Therefore,  at  9  cents  a  pound,  7 
pounds  of  sugar  will  cost  63  cents. 

2.  At  6  dollars  ?  ^eek,  what  will  8  weeks7  board  cost  ? 

3.  When  flour  is  (  dollars  a  barrel,  what  will  11  barrels 
cost? 

4.  If  Rollin  can  earn  10  dollars  in  one  month,  how  much 
can  he  earn  in  4  months  ?  in  9  months  ?  in  11  months  ? 

5.  What  will  be  the  cost  of  12  pounds  of  coffee,  at  9 
cents  a  pound  ? 

6.  At  5  dollars  a  ton,  what  will  9  tons  of  coal  cost  ? 

7.  At  4  dollars  a  yard,  what  will  8  yards  of  cloth  cost  ? 

8.  If  a  pair  of  boots  cost  5  dollars,  what  will  be  the  cost 
of  3  pairs  ?  of  6  pairs  ?  of  7  pairs  ?    of  11  pairs  1 

9.  Since  12  inches  make  a  foot,  how  many  inches  in  3  feet  ? 
in  5  feet  ?  in  7  feet  ?  in  9  feet  ?  in  12  feet  ? 

10.  At  five  cents  a  quart,  what  will  6  quarts  of  milk 
cost  ?  10  quarts  ?  11  quarts  ? 

11.  If  a  man  earn  8  dollars  in  a  week,  how  much  can  he 
earn  in  6  w;eeks  ?  in  7  weeks  ?  in  8  weeks  ?  in  9  weeks  ? 

12.  If  9  bushels  of  apples  buy  one  barrel  of  flour,  how 
many  bushels  will  be  required  to  buy  3  barrels  ?  5  barrels  ? 
7  barrels  *?  9  barrels  ? 

13.  If  4  men  can  do  a  piece  of  work  in  8  days,  how 
many  days  will  it  take  one  man  to  do  it  ? 

14.  If  7  men  can  build  a  wall  in  3  days,  how  long  will  it 
take  one  man  to  build  it  ? 

15.  If  a  barrel  of  potatoes  last  6  persons  3  weeks,  how 
many  weeks  will  it  last  one  person  ? 


MULTIPLICATION. 


PROMISCUOUS  MULTIPLICATION  TABLE. 


2  times  8  are  how  many  1 
3  times  9  are  how  many  1 
4  times  8  are  how  many  1 
7  times  5  are  how  many  ? 
9  times  4  are  how  many  ? 
6  times  3  are  how  many  ? 
4  times  9  are  how  many  1 
5  times  9  are  how  many  ? 
7  times  6  are  how  many  1 

2  times  9  are  how  many  ? 
6  times  5  are  how  many  ? 
4  times  7  are  how  many  ] 
9  times  3  are  how  many  1 
5  times  7  are  how  many  ? 
5  times  8  are  how  many  1 
9  times  5  are  how  many  1 
6  times  4  are  how  many  1 
8  times  3  are  how  many  ? 

3  times  7  are  how  many  ? 
8  times  9  are  how  many  1 
6  times  8  are  how  many  1 
5  times  6  are  how  many  ] 
7  times  3  are  how  many  1 
6  times  6  are  how  many  ? 
9  times  7  are  how  many  1 
3  times  8  are  how  many  1 
7  times  4  are  how  many  ? 

7  times  7  are  how  many  ? 
4  times  2  are  how  many  ? 
9  times  -9  are  how  many  ? 
4  times  3  are  how  many  ? 
6  times  9  are  how  many  ? 
2  times  6  are  how  many  1 
8  times  5  are  how  many  ? 
4  times  4  are  how  many  ? 
9  times  8  are  how  many  1 

8  times  7  are  how  many  ? 
5  times  4  are  how  many  1 
3  times  5  are  how  many  ? 
3  times  4  are  how  many  ? 
8  times  6  are  how  many  ? 
7  times  8  are  how  many  ? 
5  times  3  are  how  many  1 
3  times  6  are  how  many  1 
8  times  8  are  how  many  ? 

2  times  4  are  how  many  ? 
5  times  9  are  how  many  ? 
9  times  8  are  how  many  ? 
3  times  3  are  how  many  ? 
2  times  3  are  how  many  ? 
7  times  4  are  how  many  ? 
0  times  8  are  how  many  ? 
3  times  6  are  how  many  ? 
6  times  10  are  how  many  ? 

The  Multiplicand  is  the  number  to  be  taken. 
The  Multiplier  is  the  number  which  shows  how 
many  times  the  multiplicand  is  to  be  taken. 

54.  The  Factors  are  the  multiplicand  and  multiplier. 

55.  The  Sign  of  Multiplication  is  the  oblique  cross, 
X  •     It  indicates  that  the  numbers  connected  by  it  are  to 
be  multiplied  together ;  thus  9x6  =  54,  is  read  9  times  0 
equals  54. 


42  SIMPLE  NUMBERS. 

NOTES.    1.  Factors  axe  producers,  and  the  multiplicand  and  multiplier  are  called 
factors  because  they  produce  the  product. 

2.  Multiplication  is  a  short  method  of  performing  addition  when  the  numbers  to 
be  added  are  equal. 


CASE   I. 

56.  When  the  multiplier  consists  of  one  figure. 
1.  Multiply  374  by  6. 

OPERATION.  ANALYSIS.     In  this  example  it  is  re- 

quired to  take  374  six  times.     If  we 
take  the  units  of  each  order  6  times, 
Multiplicand,         374          we   shall   take    the   entire  number  6 
times.     Therefore,  writing  the  multi- 

°f  the  mul" 


Product,  2244 

tiplicand,  we   proceed  as  follows:     6 

times  4  units  are  24  units,  which  is  2  tens  and  4  units  ;  write  the 
4  units  in  the  product  in  units'  place,  and  reserve  the  2  tens  to 
add  to  the  next  product;  6  times  7  tens  are  42  tens,  and  the 
two  tens  reserved  in  the  last  product  added,  are  44  tens,  which 
is  4  hundreds  and  4  tens  ;  write  the  4  tens  in  the  product  in 
tens'  place,  and  reserve  the  4  hundreds  to  add  to  the  next  prod- 
uct; 6  times  3  hundreds  are  18  hundreds,  and  4  hundreds  add- 
ed are  22  hundreds,  which,  being  written  in  the  product  in  the 
places  of  hundreds  and  thousands,  gives,  for  the  entire  product, 
2244. 

57.  The  unit  value  of  a  number  is  not  changed  by  re- 
peating the  number.  As  the  multiplier  always  expresses 
times,  the  product  must  have  the  same  unit  value  as  the  mul 
tiplicand.  But,  since  the  product  of  any  two  numbers  will 
be  the  same,  whichever  factor  is  taken  as  a  multiplier,  either 
factor  may  be  taken  for  the  multiplier  or  multiplicand. 

NOTE.  In  multiplying,  learn  to  pronounce  the  partial  results,  as  in  addition, 
without,  naming  tin;  numbers  separately.  Thus,  in  the  lasl  example,  instead  of  say- 
ing ti  times  4  are  -4,  fi  times  7  are  42  and  2  to  carry  an.-  44.  ti  limes  3  are  18  and  4 
to  carry  are  22;  prtmouncfr  only  the  results,  24,  44,  22,  performing  the  operation* 
qaentally.  Tina  will  greatly  facilitate  the  process  of  multiplying. 


MULTIPLICATION. 


EXAMPLES   FOR   PRACTICE. 

Multiplicand, 

(2.) 

842 

(3.) 
625 

(40 
718 

Multiplier, 
Product, 

4 

6 

7 

3368 

3750 

5026 

(6.) 

4328 

(7.) 
5073 

(8.) 
1869 

8 

5 
25365 

4 

34624 

7476 

(10.) 

7186 

(11.) 
9010 

(12.) 
4079 

3 

7 

6 

21558 

63070 

24474 

(14.) 
340071 

(15.) 

760892 

2 

4 

680142 


3043568 


*  (6.) 

937 

fc 

2811 


29385 

(13.) 
6394 

£ 

51152 


9881150 


17.  Multiply  473126  by  9.  Ans. 

18.  Multiply  30789167  by  7.  Ans. 

19.  Multiply  87231420  by  8.  Ans. 

20.  What  will  be  the  cost  of  9380  bushels  of  wheat,  at  9 
shillings  a  bushel  ?  Ans.  84420  shillings. 

21.  What  will  be  the  cost  of  4738  tons  of  coal,  at  4  dol- 
lars a  ton  ?  Ans.  18952  dollars. 

22.  In  one  mile  are  5280   feet;    how  many  feet  in  8 
miles  ?  Ans.  42240  feet. 


44  SIMPLE    NUMBERS. 

CASE  II. 

58.  When  the  multiplier  consists  of  two  or  more 
figures. 

1.  Multiply  746  by  23. 

OPERATION.  ANALYSIS.   Writ- 

Multiplicand,  746  ing    the    multipli- 

cand  and  multipli- 
f  times  the  mul-         er  as  in  Case  I,  we 


figure  in  the  mul- 


multiplier,  precisely  as  in  Case  I.  We  then  multiply  by  the  2  tens. 
2  tens  times  6  units,  or  6  times  2  tens,  are  12  tens,  equal  to  1 
hundred,  and  2  tens  ;  we  place  the  2  tens  under  the  tens  figure 
in  the  product  already  obtained,  and  add  the  1  hundred  to  the 
next  hundreds  produced.  2  tens  times  4  tens  are  8  hundreds, 
and  the  1  hundred  of  the  last  product  added  are  9  hundreds  ; 
we  write  the  the  9  in  hundreds'  place  in  the  product.  2  tens 
times  7  hundreds  are  14  thousands,  equal  to  1  ten  thousand  and  4 
thousands,  which  we  write  in  their  appropriate  places  in  the 
the  product.  Then  adding  the  two  products,  we  have  the  en 
tire  product,  17158. 

Hence  we  deduce  the  following  general 
II  OLE.     I.    Write  the  multiplier  under  the  multiplicand, 
placing  units  of  the  same  order  under  each  other. 

II.  Multiply  the  multiplicand  by  each  figure  of  the  multi- 
plier successively,  beginning  with  the  unit  figure,  and  write 
the  first  figure  of  each  partial  product  under  the  figure  of 
the  multiplier  used,  writing  down  and  carrying  as  in  addi- 
tion. 

III.  If  there  are  partial  products,  add  them,  and  their 
mm  will  be  the  product  required. 


MULTIPLICATION. 


45 


PROOF.  Multiply  the  multiplier  by  the  multiplicand,  and 
if  the  product  is  the  same  as  the  first  result,  the  work  is 
correct. 

NOTH.    When  the  multiplier  contains  two  or  more  figures,  the  several  results  ob- 
tained by  multiplying  by  each  figure  are  called  partial  products. 


EXAMPLES   FOB   PRACTICE. 

(1.)                       (20 

(3.) 

34732                 56784 

34075 

14                       24 

36 

138928                227136 

204450 

34732                113568 

102225 

486248              1362816 

1226700 

4.  Multiply    177242  by  19. 

Am.    3367598. 

5.  Multiply  1429689  by  55. 

6.  Multiply    364111  by  56. 

Ant.  20390216. 

7.  Multiply       78540  by  95. 

An*.     7461300. 

8.  Multiply         6555  by  39. 

An*.       255645. 

9.  Multiply      76419  by  17. 

Ans.     1299123. 

10.  Multiply       26517  by  45. 

Ans.     1193265. 

11.  Multiply    108336  by  58. 

Ans.     6283488. 

12.  Multiply  209402  by  72. 

Ans.    15076944. 

13.  Multiply  342516  by  56. 

Ans.    19180896. 

14.  Multiply  764131  by  48. 

Ans.    36678288. 

15.  There  are  52  weeks  in  a  year ;  how  many  weeks  in 
1861  years?  Ans.  $6772  weeks. 

16.  An  army  of  5746  men  having  plundered  a  city,  took 
so  much  money  that  each  man  received  37   dollars ;  how 
much  money  was  taken  ?  Ans.  212602  dollars. 

17.  If  it  cost  47346  dollars  to  build  one  mile  of  railroad, 
how  much  will  it  cost  to  build  76  miles  ? 

Ans.  3598296  dollars. 


SIMPLE    NUMBERS. 


190784 
190784 
47696 

6868224 


(19.) 
560341 
304 

2241364 
1681023 

170343664 


(20.) 
243042 
265 

1215210 
1458252 

486084 

64406130 


21.  Multiply  45678  by  333.  Ans.         15210774. 

22.  Multiply  202842  by  342.         Ans.         69371964.' 

23.  Multiply  9636799  by  489.       Ans.     4712394711. 

24.  Multiply  3064125  by  807.       Ans.     2472748875. 

25.  Multiply  5610327  by  2034. 

26.  Multiply  1900731  by  4006.    Ans.     7614328386. 

27.  A  gentleman  bought  307  horses  for  shipping,  at  the 
rate  of  105  dollars  each ;   how  much  did  he  pay  for  the 
whole  ? 

28.  What  would  be  the  value  of  976  shares  of  railroad 
stock,  at  98  dollars  a  share  ?  Ans.  95648  dollars. 

29.  A  man  bought  48  building  lots,  at  1236  dollars  each; 
how  much  did  they  all  cost  him  1       Ans.  59328  dollars. 

30.  How  many  yards  of  broadcloth  in  487  pieces,  each 
piece  containg  37  yards  ?  Ans.  18019  yards. 

31.  If  it  require  135  tons  of  iron  for  one  mile  of  railroad, 
how  many  tons  will  be  required  for  196  miles  ? 

Ans.  26460. 

32.  How  many  oranges  in  356  boxes,  each  box  contain- 
ing 264  oranges  ?  Ans.  93984. 

33.  If  it  require  6894  shingles  for  the  roof  of  a  house, 
how  many  shingles  will  be  required  for  19  such  houses  ? 


MULTIPLICATION.  47 

34.  37896X149  =how  many  ?        A-ns.         5646504. 

35.  8567X462  =how  many1?          Ans.         3957954. 

36.  6793X842  =how  many  ?          Ans.         5719706. 

37.  674200 X 2104  =how  many?    Ans.  1418516800. 

38.  15607X3094  =how  many?      Ans.      48288058. 

39.  83209X4004  =how  many  ? 

40.  Multiply  31416  by  175. 

41.  Multiply  40930  by  779.  Ans.  31884470. 

42.  Multiply  4567  by  9009.  Ans.  41144103. 

43.  Multiply  7071  by  556.  Ans.    3931476. 

44.  Multiply  291042  by  125.  Ans.  36380250. 

45.  Multiply  54001  by  5009. 

46.  Multiply  twelve  thousand  thirteen,  by  twelve  hun- 
dred four.  Ans.  14463652. 

47.  Multiply  thirty-seven  thousand  seven  hundred  ninety- 
six,  by  four  hundred  eight. 

48.  Multiply  one  million  two  hundred  forty-six  thousand 
eight  hnndred  fifty-three,  by  nine  thousand  seven. 

-4ns.  11230404971. 

49.  What  will  be  the  cost  of  building  128  miles  of  rail- 
road, at  6375  dollars  per  mile  1         Ans.  816000  dollars. 

50.  A  crop  of  cotton  was  put  up  in  126  bales,  each  bale 
containing  572  pounds ;  what  was  the  weight  of  the  entire 
crop  ?  Ans.  72072  pounds. 

51.  Two  towns,  243  miles  apart,  are  to  be  connected  by  a 
railroad,  at  a  cost  of  39760  dollars  a  mile ;  how  much  will 
be  the  entire  cost  of  ^he  road  ?         Ans.  9661680  dollars. 

52.  Allowing  an  acre  of  land  to  produce  105  bushels,  how 
much  would  246  acres  produce  ?          Ans.  25830  bushels. 

53.  If  a  garrison  of  soldiers  consume  5789   pounds  of 
bread  a  day,  how  much  will  they  consume  in  287  days  1 

Ans.  1661443  pounds 


48  SIMPLE   NUMBERS. 

CONTRACTIONS. 
CASE     I. 

59.  When  the  multiplier  is  a  composite  num- 
ber. 

A  Composite  Number  is  one  that  may  be  produced  by 
multiplying  together  two  or  more  numbers ;  thus,  18  is  a 
composite  number,  since  6x3=18 ;  or;  9x2=18  ;  or,  3X 
3X2=18. 

@®.  The  Component  Factors  of  a  number  are  the  sev- 
eral numbers  which,  multiplied  together,  produce  the  given 
number ;  thus,  the  component  factors  of  20  are  10  and  2, 
(10X2=20);  or,  4  and  5,  (4x5=20)  ;  or,  2  and  2  and 
5,  (2X2X5=20). 

XOTE. — The  pupil  must  not  confound  the  factors  with  the  parts  of  a  number. 
Thus,  the/acto?-s  of  which  twelve  is  composed,  are  4  and  3,  (4X3=12)  ;  while  the 
parts  of  which  12  is  composed  are  8  and  4,  (8+4=12),  or  10  and  2,  (10+2=12) 
The  factors  are  multiplied,  while  the  parts  are  added,  to  produce  the  number. 

I.  What  will  32  horses  cost,  at  174  dollars  apiece  ? 

OPERATION.  ANALYSIS.    The  fac- 

Muitipiicand       174  cost  of  1  horse.         tors  of  32  are  4  and 
1st  factor,  4  8.     If  we  multiply  the 

cost  of  1  horse  by  4, 

696  cost  of  4  horses.       we  obtain  the  cost  of 
2d  factor,  8  4  horses;  and  by  mul- 

tiplying the  cost  of  4 
product,         5568  cost  of  32  horses.       horses  by  8j  we  obtain 

the  cost  of  8  times  4  horses,  or  32  horses,  the  number  bought 
61.  Hence  we  have  the  following 
RULE.     I.  Separate  the  composite  number  into  two  or 
more  factors. 

II.  Multiply  the  multiplicand  "by  one  of  these  factors,  and 
that  product  l)y.  another,  and  so  on  until  all  the  factors  have 
Lecn  used)  the  last  product  will  be  the  product  required. 

NOTE.  The  product  of  any  nupiber  of  factors  will  be  the  same  in  whatever  order 
they  arc  multiplied.  Thus,  4X-';X5— 60,  and  5X4X3-CO. 


{ 
MULTIPLICATION.  49 


EXAMPLES   TOR   PRACTICE. 

1.  Multiply  521  by  16=4x4.  !       Am.  8336. 

2.  Multiply  4350  by  25  =  5x5.  Ans.  108750. 

3.  Multiply  10709  by  36=6x6.  Ans.  385524. 

4.  Multiply  21700  by  27=3x9.  Am.  585900. 

5.  Multiply  783473  by  42=6x7.  jAns.  32905866. 

6.  Multiply  764131,  by  48=6X8.  lAns.  36678288. 

7.  Multiply  40567  by  96=8x12.  |  Ans.  3894432. 

8.  Multiply  182642  by  120=4x5Xf 

\Ans.  21917040. 

9.  Multiply  20704  by  84=3X4X7.;   Ans.  1739136. 

10.  Multiply  564120  by  140=4X5X7. 

I  Ans.  78976800. 

11.  What  will  56  acres  of  land  cost,  at  147  dollars  an 
acre?  Ans.  8232  dollars. 

12.  What  will  75  yoke  of  cattle  cf>st,  at  184  dollars  a 
yoke  ?  Ans.  13800  dollars. 

13.  If  a  ship  sail  380  miles  a  day,  how  far  will  she  sail 
in  45  days  1  Ans.  17100  miles. 

14.  What  is  the  value  of  3426  pounds  of  butter,  at  18 
cents  a  pound  ?  tAns.  61668  cents. 

15.  What  would  be  the  cost  of  125  horses,  at  208  dollar 
each  ?  Ans.  26000  dollars. 

16.  What  would  be  the  value  of  1^42  acres  of  land,  at 
28  dollars  an  acre  ? 

17.  What  will  be  the  cost  of  28  pieces  of  broadcloth,  each 


piece  containing  42  yards,  at  4  dollar 


Ans.  4704  dollars. 

18.  What  will  be  the  cost  of  16  sa<fks  of  coffee,  each  sack 
containing  7r>  pounds,  at  9  cents  a  pound  ? 

'  Ans.  10800  cents. 


a  yard 


50  SIMPLE    NUMBERS. 

CASE  II. 

62.  When  the  multiplier  is  10,  *100,  1000,  &c. 

If  we  annex  a  cipher  to  the  multiplicand,  each  figure  is 
removed  one  place  toward  the  left,  and  consequently  the 
value  of  the  whole  numher  is  increased  ten  fold.  If  two 
ciphers  are  annexed,  each  figure  is  removed  two  places  to- 
ward the  left,  and  the  value  of  the  numher  is  increased  one 
hundred  fold  j  and  every  additional  cipher  increases  the 
value  tenfold. 

Hence  the  following 

RULE.  Annex  as  many  ciphers  •  to  the  multiplicand  as 
there  are  ciphers  in  the  multiplier ;  the  number  so  formed 
will  be  the  product  required. 

EXAMPLES    FOR   PRACTICE. 

1.  Multiply  246  by  10.  Ans.  2460. 

2.  Multiply  97  hy  100.  Ans.          9700. 

3.  Multiply  1476  hy  1000.  Ans.     1476000. 

4.  Multiply  7361  by  10000.  Ans.  73610000. 

5.  At  47  dollars  an  acre,  what  will  10  acres  of  land  cost  ? 

Ans.  470  dollars. 

6  What  will  be  the  cost  of  100  horses,  at  95  dollars  a 
head  ?  Ans.  9500  dollars. 

7.  What  will  be  the  cost  of  1000  fruit  trees,  at  18  cents 
apiece?  Ans.  18000  cents. 

8.  If  one  acre  of  land  produce  28  bushels  of  wheat, 
how  many  bushels  will  100  acres  produce  ?     Ans.  2800. 

9.  If  a  man  save  386  dollars  a  year,  how  much  will  he 
save  in  10  years  ?  Ans.  3860  dollars. 

10.  If  the  freight  on  a  barrel  of  flour  from  Chicago  to 
New  York  be  47  cents,  how  much  will  it  be  on  100000  bar- 
rels ?  Ans.  4700000  cents. 


MULTIPLICATION.  61 

CASE  III. 

63.  When  there  are  ciphers  at  the  right  hand  of 
one  or  both  of  the  factors. 

1.  Multiply  7200  by  40. 

OPERATION.  ANALYSIS.    The  multiplicand,  fac- 

Muibpiicand,    7200  tored,  is  equal  to  72  X  100 ;  the  mul- 

Moltipiier,          40  tiplier,  factored,  is  equal  to  4  x  10 

and  as  these  factors  taken  in  any 
order  will  give  the  same  product, 

we  first  multiply  72  by  4,  then  this  product  by  100  by  annex- 
ing two  ciphers,  and  this  product  by  10  by  annexing  one  a 
pher.     Hence,  the  following 

RULE.  Multiply  the  significant  figures  of  the  multipli- 
cand by  those  of  the  multiplier,  and  to  the  product  annex  as 
many  ciphers  as  there  are  ciphers  on  the  right  of  either  or 
loth  factors. 

EXAMPLES    FOR   PRACTICE. 

(1.)  (2.)  (3.) 

Multiply    3900  1760  37200 

By  8000  3500  730000 

31200000  880  1116 

528  2604 


6160000  271560000CO 

4.  Multiply  7030  by  164000.         Ans.     1152920000. 

5.  Multiply  27600  by  48000.         Ans.     1324800000. 

6.  Multiply  403700  by  30200.       Ans.  12191740000. 

7.  At  150  dollars  an  acre,  what  will  be  the  cost  of  500 
acres  ol  land  1  Ans.  75000  dollars. 

8.  What  will  be  the  freight  on  4000  barrels  of  flour,  at 
50  cents  a  barrel  1  Ans.  200000  cents. 

9.  If  there  are  560  shingles  in  a  bunch,  how  many  shin- 
gles in  26ITO  bunches  ?  Ans.  14560000. 


52  SIMPLE   NUMBERS. 

EXAMPLES     COMBINING     ADDITION,     SUBTRACTION,     AND 
MULTIPLICATION. 

1.  Bought  9  cords  of  wood  at  3  dollars  a  cord,  and  15 
tons  of  coal  at  5  dollars  a  ton ;  what  was  the  cost  of  the 
wood  and  coal  ?  Ans.  102  dollars. 

2.  A  grocer  bought  6  tubs  of  butter,  each  containing  64 
pounds,  at  14  cents  a  pound;  and  4  cheeses,  each  weighing 
42  pounds,  at  8  cents  a  pound ;  how  much  was  the  cost  of 
the  butter  and  cheese  ?  Ans.  6720  cents. 

3.  If  a  clerk  receive  540  dollars  a  year  salary,  and  pay 
180  dollars  for  board,  116  dollars  for  clothing,  58  dollars 
for  books,  and  75  dollars  for  other  expenses,  how  much  will 
he  have  left  at  the  close  of  the  year  ?     Ans.  Ill  dollars. 

4.  A  farmer  having  2150  dollars,  bought  536  sheep  at 
2  dollars  a  head,  and  26  cows  at  23  dollars  a  head ;  how 
much  money  had  he  left  ?  Ans.  480  dollars. 

5.  A  man  sold  three  horses ;  for  the  first  he  received  275 
dollars,  for  the  second  87  dollars  less  than  for  the  first,  and 
for  the  third  as  much  as  for  the  other  two ;  how  much  did 
he  receive  for  the  third  ?  Ans.  463  dollars. 

6.  Bought  76  hogs,  each  weighing  416  pounds,  at  7 
cents  a  pound,  and  sold  the  same  at  9  cents  a  pound ;  how 
much  was  gained  ?  Ans.  63232  cents. 

7.  A  man  bought  14  cows  at  26  dollars  each,  4  horses 
at  112  dollars  each,  and  125  sheep  at  3  dollars  each  ]  he  sold 
the  whole  for  1237  dollars ;  did  he  gain  or  lose,  and  how 
much?  Ans.  Gained  50  dollars. 

8.  B  has  174  sheep,  C  has  three  times  as  many  lacking 
98,  and  D  has  as  many  as  B  and  C  together ;  how  many 
eheep  has  D 1  Ans.  598. 

9.  There  are  36  tubs  of   butter,  each    weighing    108 
pounds ;  the  tubs  which  contain  the  butter,  each  weigh  19 


PKOMISCUOUS  EXAMPLES.  63 

pounds  j  how  much  is  the  weight  of  the  butter  without  the 
tubs  ?  Ans.  3204  pounds. 

10  A  man  paid  for  building  a  house  2376  dollars,  and  for 
his  farm  4  times  as  much  lacking  970  dollars ;  how  much 
did  he  pay  for  both  ? 

11.  A  merchant  bought  9  hogsheads  of  sugar  at  32  dol- 
lars a  hogshead,  and  sold  it  for  40  dollars  a  hogshead ;  how 
much  did  he  gain  ?  Ans.  72  dollars. 

12.  Bought  360  barrels  of  flour  for  2340  dollars,  and  sold 
the  same  at  8  dollars  a  barrel ;  how  much  was  gained  by 
the  bargain  ?  Ans.  540  dollars. 

13.  A  farmer  sold  462  bushels  of  wheat  at  2  dollars  a 
bushel,  for  which  he  received  75  barrels  of  flour  at  9  dol- 
lars a  barrel,  and  the  balance  in  money ;  how  much  money 
did  he  receive  ?  Ans.  249  dollars. 

14.  Two  persons  start  from  the  same  point,  and  travel  in 
opposite  directions ;  one  travels  at  the  rate  of  28  miles  a 
day,  the  other  at  the  rate  of  37  miles  a  day ;  how  far  apart 
will  they  be  in  6  days  ?  Ans.  390  miles. 

15.  If  a  man  buy  40  acres  of  land  at  35  dollars  an  acre, 
and  56  acres  at  29  dollars  an  acre,  and  sell  the  whole  for 
32  dollars  an  acre,  how  much  does  he  gain  or  lose  ? 

Ans.  Gains  48  dollars. 

16.  In  an  orchard,  76  apple  trees  yield  18  bushels  of  ap- 
ples each,  and  27  others  yield  21  bushels  each ;  how  much 
would  the  apples  be  worth,  at  30  cents  a  bushel  ? 

Ans.  58050  cents. 

17.  A  man  bought  two  farms,  one  of  136  acres  at  28 
dollars  an  acre,  and  another  of  140  acres  at  33  dollars  an 
acre ;  he  paid  at  one  time  4000  dollars,  and  at  another  time 
1875  dollars ;  how  much  remained  unpaid  ? 

Ans.  2553  dollars. 


64:  SIMPLE    NUMBERS. 

DIVISION. 

G4:.  Division  is  the  process  of  finding  how  many  times 
one  number  is  contained  in  another. 

G«5.  The  Quotient  is  the  result  obtained,  and  shows  how 
many  times  the  divisor  is  contained  in  the  dividend. 

DIVISION   TABLE. 


1    in    2   2    times 

2    in    4    2    times 

8    in    6   2    times 

1    in    3    3    times 

2    in    63    times 

3    in    9    3    times 

1    in    4    4    times 

2    in    8    4    times 

8    in  12    4    times 

1    in    5    5    times 

2    in  10    5    times 

3    in  15   5    times 

1    in    6    6    times 

2    in  12    6    times 

3    in  18   6    times 

1    in    7    7    times 

2    in  14    7    times 

,3    in  21    7    times 

1    in    8    8    times 

2    in  16    8    times 

8    in  24    8    times 

1    in    9    9    times 

2    in  18    9    times 

8    in  27    9    times 

4   in      8    2  times 

5    in    10    2  times 

6    in    12    2  times 

4   in    12    3  times 

5    in    15    3-  times 

6    in    18    3  times 

4    in    16    4  times 

B    in    20    4  times 

6    in    24    4  timea 

4    in    20    5  times 

6    in    25    5  times 

6    in    30    5  times 

4    in    24    6  times 

5    in   30    6  times 

6    in    36    6  times 

4   in    28    7  times 

5    in   35    7  times 

6    in    42    7  times 

4    in    32    8  times 

5    in   40    8  times 

6    in    48    8  times 

4   in    36    9  times 

5    in    45    9  times 

6    in    54    9  times 

7    in   14    2  times 

8    in    16    2  times 

9    in    18    2  times 

7    in    21    3  times 

8    in    24    3  times 

9    in    27    3  times 

7    in    28    4  times 

8    in    32    4  times 

9    in    36   4  times 

7    in   35    6  times 

8    in    40    5  times 

9    in    45   5  times 

7    in   42    6  times 

8    in    48    6  times 

9    in    54    6  times 

7    in   49    7  times 

8    in    56    7  times 

9    in    63    7  times 

7    in    66    8  times 

8    in    64    8  times 

9    in    V2    8  times 

7    in    63    9  times 

8    in    72    9  times 

9    in    81    9  times 

10    in    20    2  times 

11    in    22    2  times 

12    in    24    2  times 

10    in    30    3  times 

11    in   S3    3  times 

12    in    36    3  times 

10    in    40    4  times 

11     in    44    4  times 

12    in    48    4  times 

10    in     60     5  times 

11    in   55    5  times 

12   in    60    5  times 

10    in    60    6  times 

11    in    66    6  times 

12    in    72    6  times 

10    In    70    7  times 

11    in    77     7  times 

12    in    84    7  times 

10    in    80    8  times 

11    in   88    8  times 

12    in    96    8  times 

10    in    90    9  times 

11    in    99    9  times 

12    in  108    9  times 

DIVISION.  56 

MENTAL   EXERCISES. 

1.  How  many  barrels  of  flour;  at  6  dollars  a  barrel  can  be 
bought  for  30  dollars  ? 

ANALYSIS.     Since  6  dollars  will  buy  one  barrel  of  flour,  30  dol- 
lars will  buy  as  many  barrels  as  6  dollars,  the  price  of  one  barrel, 
Is  contained  times  in  30  dollars,  which  is  5  times.    Therefore,  at 
dollars  a  barrel,  5  barrels  of  flour  can  be  bought  for  30  dollars. 

2.  How  many  oranges,  at  4  cents  apiece,  can  be  bought 
for  28  cents  ? 

3.  How  many  tons  of  coal;  at  5  dollars  a  ton,  can  be 
bought  for  35  dollars  ? 

4.  When  lard  is  7  cents  a  pound,  how  many  pounds  can 
be  bought  for  49  cents  ?  for  63  cents  ?  for  84  cents  ? 

5.  If  a  man  travel  48  miles  in  6  hou^how  far  does  he 
travel  in  one  hour  ? 

6.  At  3  cents  apiece,  how  many  lemons  can   be  bought 
for  24  cents  ?  for  30  cents  ?  for  36  cents  ? 

7.  If  you  give  55  cents  to  5  beggars,  how  many  cents  do 
you  give  to  each  ? 

8.  If  a  man  build  42  rods  of  wall  in  7  days,  how  many 
rods  can  he  build  in  1  day  ? 

9.  At  4  dollars  a  cord,  how  many  cords  of  wood  can  be 
bought  for  20  dollars  ?  for  28  dollars  ?  for  32  dollars  ? 

10.  A  farmer  paid  33  dollars  for  some  sheep,  at  3  dollars 
apiece ;  how  many  did  he  buy  ? 

11.  At  7  cents  a  pound,  how  many  pounds  of  sugar  can 
be  bought  for  63  cents  ?  for  84  cents  ? 

12.  If  a  man  spend  5  cents  a  day  for  cigars,  how  many 
days  will    50  cents  last  him  ?  60  cents  1 

13.  At  12  cents  a  pound,  how  many  pounds  of  coffee  can 
be  bought  for  48  cents?  for  72  cents?  for  96  cents?  for 
120  cents  ? 


56 


SIMPLE   NUMBERS. 


PROMISCUOUS   D 

6  in  36,  how  many  times  ? 
7  in  42,  how  many  times  1 
9  in  81,  how  many  times  1 
5  in  35,  how  many  times  ? 
8  in  72,  how  many  times  1 
9  in  27,  how  many  times  1 
4  in  20,  how  many  times  ? 
6  in  54,  how  many  times  ? 

I  VISION   TABLE. 

9  in  63,  how  many  times  ? 
6  in  12,  how  many  times  1 
7  in  28,  how  many  times  ? 
4  in  16,  how  many  times  ? 
7  in  49,  how  many  times  ? 
4  in  36,  how  many  times  ? 
8  in  64,  how  many  times  ? 
8  in  40,  how  many  times  ? 

8  in  32,  how  many  times  ? 
5  in  45,  how  many  times  ? 
6  in  42,  how  many  times  ? 
8  in  56,  how  many  times  1 
7  in  63,  how  many  times  ? 
3  in  27,  how  many  times  ? 
7  in  21,  how  JM^J  times  1 
8  in  16,  how  many  times  1 

4  in  28,  how  many  times  ? 
8  in  32,  how  many  times  ? 
6  in  48,  how  many  times  ? 
9  in  45,  how  many  times  ? 
8  in  48,  how  many  times  ? 
7  in  56,  how  many  times  ? 
3  in  21,  how  many  times  ? 
6  in  54,  how  many  times  ? 

4  in  12,  how  many  times  ? 
7  in  35,  how  many  times  ? 

5  in  10,  how  many  times  ? 
7  in  14,  how  many  times  ? 

4  in  24,  how  many  times  1 

5  in  30,  how  many  times  ? 
9  in  36,  how  many  times  ? 

6  in  30,  how  many  times  ? 


2  in  16,  how  many  times  ? 

4  in  32,  how  many  times  ? 
6  in  24,  how  many  times  ? 
9  in  72,  how  many  times  ? 

5  in  10,  how  many  times  ? 

4  in    8,  how  many  times  ? 

5  in  20,  how  many  times  ? 
2  in  10,  how  many  times  ? 


66.  The  Dividend  is  the  number  to  be  divided. 

67.  The  Divisor  is  the  number  to  divide  by. 

68.  The  Sign  of  Division  is  a  short  horizontal  line, 
with  a  point  above  and  one  below,  -+-.     It  indicates  that  the 
number  before  it  is  to  be  divided  by  the  number  after  it. 
Thus,  20  .-*-  4  =  5,  is  read,  20  divided  by  4  is  equal  to  5. 

Division  is  also  expressed  by  writing  the  dividend  abovef 
and  the  divisor  below,  a  short  horizontal  line ; 

12 
Thus,    ~-=  4,  shows  that  12  divided  by  3  equals  4. 


DIVISION.  57 

CASE   I. 

69.  When  the  divisor  consists  of  one  figure. 

1.  How  many  times  is  4  contained  in  848  ? 

OPERATION.  ANALYSIS.     After  writing  the  divisor 

Dividend,         On  the  left  of  the  dividend,  with  a  line 

DiTisor>  between  them,  we  begin  at  the  left  hand 

oi  0          an(*  Sa7 :  ^  is  contained  in  8,  2  times, 

and  as  8  in  the  dividend  is  hundreds, 

the  2  in  the  quotient  must  be  hundreds ;  we  therefore  write  2 
in  hundreds'  place  under  the  figure  divided.  4  is  contained  in 
4, 1  time,  and  since  4  denotes  tens,  we  write  1  under  it  :a  tens' 
place.  4  in  8,  2  times,  and  since  8  is  units,  we  write  2  in  units' 
place  under  it,  and  we  have  212  for  the  entire  quotient. 

EXAMPLES'  FOR   PRACTICE. 

(2.)  (3.)  (4.) 

Wrisor,       3)936   Dividend,  2)4862  4)48844 

312    Quotient.  2431  12211 

5.  Divide  9963  by  3.  Ans.     3321. 

6.  Divide  5555  by  5.  Ans.    1111. 

7.  Divide  68242  by  2.  Ans.  34121. 

8.  Divide  66666  by  6. 

When  the  divisor  is  not  contained  in  the  first  figure  of 
the  dividend,  we  find  how  many  times  it  is  contained  in  the 
first  two  figures. 

9.  How  many  times  is  4  contained  in  2884  ? 
OPERATION.  ANALYSIS.     As  we  cannot  divide  2  by  4, 

4)2884         we  say  4  is  contained  in  28,  7  times,  and 

write  the  7  in  hundreds'  place;  then  4  is 

721          contained  in  8,  2  times,  which  we  write  m 

tens'  place  under  the  figure  divided ;  and  4  is  contained  in  4,  1 

time,  which  we  write  in  units'  place  in  the  quotient,  and  we 

have  the  entire  quotient,  721. 


58  SIMPLE  NUMBERS. 

EXAMPLES   FOR   PRACTICE. 

(10.)         (11.)  (12.) 

3)2469       5)3055        2)148624 

823  611     ^      74312 

13.  Divide  4266  by  6.  Ans.     711. 

14.  Divide  36488  by  4.  Ans.  9122. 

15.  Divide  72999  by  9.  Ans.  8111. 

16.  Divide  21777  by  7. 

After  obtaining  the  first  figure  of  the  quotient,  if  the  di- 
visor is  not  contained  in  any  figure  of  the  dividend,  place  a 
cipher  in  the  quotient,  and  prefix  this  figure  to  the  next 
one  of  the  dividend. 

NOTE.    To  prefix  means  to  place  before,  or  at  the  left  hand. 

17.  How  many  times  is  6  contained  in  1824  ? 
OPERATION.  ANALYSIS.     Beginning  as  in  the  last  ex- 

6  )  1824  amples,  we  say,  6  is  contained  in  18,  3  times 
which  we  write  in  hundreds'  place  in  the 
quotient ;  then  6  is  contained  in  2  no  times, 
BO  we  write  a  cipher  (0)  in  tens'  place  in  the  quotient,  and  pre- 
fixing the  2  to  the  4,  we  say  6  is  contained  in  24,  4  times,  which 
we  write  in  units'  place,  and  we  have  304  for  the  entire  quo- 
tient. 

EXAMPLES   FOR  PRACTICE. 

(18.)  (19.)  (20.) 

4)3228  7)28357  3)912246 

807  4051  304082 

21.  Divide  40525  bj  5.  Ans.     8105. 

22.  Divide  36426  by  6.  Ans.     6071. 

23.  Divide  184210  by  2.  Ans.  92105. 

24.  Divide  85688  by  8.  Ans.  10711. 

25.  Divide  273615  by  3.  Ans.  91205. 


DIVISION.  69 

After  dividing  any  figure  of  the  dividend,  if  there  be  a 
remainder,  prefix  it  mentally,  to  the  next  figure  of  the  divi- 
dend, and  then  divide  this  number  as  before. 

31.  How  many  times  is  4  contained  in  943  ? 

OPERATION.  ANALYSIS.     Here  4  is  contained  in 

4 )  943  9,  2  time?,  and  there  is  1  remainder, 

which  we  prefix  mentally  to  the  next 
235  ...  3  Rem.  figure,  4,  and  say  4  is  contained  in  14, 
3  times,  and  a  remainder  of  2,  which  we  prefix  to  3,  and  say,  4 
is  contained  in  23,  5  times,  and  a  remainder  of  3.  This  3  which 
is  left  after  performing  the  last  division  should  be  divided  by  the 
divisor  4  ;  but  the  method  of  doing  it  cannot  be  explained  here, 
and  so  we  merely  indicate  the  division  by  placing  the  divisor 
under  it ;  thus,  f.  The  entire  quotient  is  written  235 1,  which 
may  be  read,  two  hundred  thirty-five  and  three  divided,  ly  four, 
or,  two  hundred  thirty-five  and  a  remainder  oj  three. 

NOTE.  When  the  process  of  dividing  is  performed  mentally,  and  the  results  only 
are  written,  as  in  the  preceeding  examples,  the  operation  is  termed  Sliort  Division. 

From  the  foregoing  examples  and  illustrations,  we  deduce 
the  following 

RULE.  I.  Write  the  divisor  at  the  left  of  the  dividend, 
with  a  line  between  them. 

II.  Beginning  at  the  left  hand,  divide  each  figure  of  the 
dividend  by  the  divisor ,  and  write  the  result  under  the  divi- 
dend. 

III.  If  there  be  a  remainder  after  dividing  any  figure, 
regard  it  as  prefixed  to  the  figure  of  the  next  lower  order  in 
the  dividend,  and  divide  as  before. 

IV.  Should  any  figure  or  part  of  the  dividend  be  less 
than  the  divisor,  write  a  cipher  in  the  quotient,  and  prefix 
the  number  to  the  figure  of  the  next  lower  order  in  the  divi- 
dend>  and  divide  as  before. 

V.  If  there  be  a  remainder  after  dividing  the  last  figure, 
place  it  over  the  divisor  at  the  right  hand  of  the  quotient. 


60 


SIMPLE  NUMBEKS. 


PROOF.  Multiply  the  divisor  and  quotient  together,  and 
to  the  product  add  the  remainder,  if  any ;  if  the  result  be 
equal  to  the  dividend,  the  work  is  correct. 

NOTES.  1.  This  method  of  proof  depends  on  the  fact  that  division  is  the  revers* 
of  multiplication.  The  dividend  answers  to  the  product,  the  divisor  to  one  of  tha 
factors,  and  the  quotient  to  the  other  factor. 

2.  In  multiplication  the  two  factors  arc  given,  to  find  the  product :  in  division, 
the  product  and  one  of  the  factors  are  given,  to  find  the  other  factor. 

EXAMPLES  FOR  PRACTICE. 

1.  Divide  8430  by  6. 

OPERATION.  PROOF 

Divisor.    6)8430   Dividend.  1405   Quotient. 

1405    Quotient 


(20 
5)730490 

146098 


5.  Divide 

6.  Divide 

7.  Divide 

8.  Divide 

9.  Divide 

10.  Divide 

11.  Divide 

12.  Divide 

13.  Divide 

14.  Divide 

15.  Divide 

16.  Divide 

17.  Divide 


(3.) 

7)510384 


72912 


87647  by  7. 
94328  by  8. 
43272  by  9. 
377424  by  6. 
975216  by  8. 
46375028  by  7. 
4763025  by  9. 
42005607  by  7. 
72000450  by  9. 
97440643  by  8. 
65706313  by  9. 
3627089  by  6. 
4704091  by  7. 


6   Divisor. 
8430    Dividend. 


(40 
8)6003424 

750428 

Quotients. 

12521. 
11791. 

4808. 
62904. 

121902. 
6625004. 

529225. 
6000801. 
8000050. 
12180801. 
7300701J. 

604514JJ. 

672013. 


DIVISION.  61 

18.  Divide  16344  dollars  equally  among  6  men;  how 
much  will  each  man  receive  ?  Ans.  2724  dollars. 

19.  How  many  barrels  of  flour,  at  7  dollars  a  barrel,  can 
be  bought  for  87605  dollars  ?  Ans.  12515  barrels. 

20.  In  one  week  there  are  7  days ;  how  many  weeks  in 
23044  days  ?  Ans.  3292  weeks. 

21.  If  5  bushels  of  wheat  make  1  barrel  of  flour,  how 
many  barrels  of  flour  can  be  made  from  314670  bushels  ? 

Ans.  62934  barrels. 

22.  By  reading  9  pages  a  day,  how  many  days  will  be  re- 
quired to  read  a  book  through  which  contains  1161  pages? 

Ans.  129  days. 

23.  At  4  dollars  a  yard,  how  many  yards  of  broadcloth 
can  be  bought  for  1372  dollars  1  Ans.  343  yards. 

24.  If  a  stage  goes  at  the  rate  of  8  miles  an  hour,  how 
long  will  it  be  in  going  1560  miles  ?         Ans,  195  hours. 

25.  There  are  3  feet  in  1  yard;   how  many  yards  in 
206175  feet1?  Ans.  68725  yards. 

26.  Five  partners  share  equally  the  loss  of  a  ship  and  car- 
go, valued  at  760315  dollars ;    how  much  is  each  one's 
share  of  the  loss  ?  Ans.  152063  dollars. 

27.  If  a  township  of   64000  acres  be  divided  equally 
among  8  persons,  how  many  acres  will  each  receive  ? 

Ans.  8000  acres. 

28.  A  miller  wishes  to  put  36312  bushels  of  grain  into  6 
bins  of  equal  size ;  how  many  bushels  must  each  bin  con- 
tain ?  Ans.  6052  bushels. 

29.  How  many  steps  of  3  feet  each  would  a  man  take  in 
walking  a  mile,  or  5280  feet  ?  Ans.  1760  steps. 

30.  A  gentleman  left  his  estate,  worth  36105  dollars,  to 
be  shared  equally  by  his  wife  and  4  children ;  how  much 
did  each  receive  ?  Ans.  7221  dollars. 


62  SIMPLE   NUMBERS. 

CASE  II. 

7O.  "When  the  divisor  consists  of  two  or  more 
figures. 

NOTE.    To  illustrate  more  clearly  the  method  of  operation,  we  mil  first  take  ao 
example  usually  performed  by  Short  Division. 

1.  How  many  times  is  4  contained  in  1504  ? 
OPERATION.  ANALYSIS.    First.   We  find  how  many  times 

4)1504(376         ^e  Divisor  ^>  *s  contained  in  15,  the  first  par- 
-•  2  tial  dividend,  which  we  find  to  be  3  times* 

and  a   remainder.      We  place  this  quotient 

30  figure  at  the  right  hand  of  the  dividend,  with 

28  a  line  between  them.     Second.     To  find  the 

remainder,  we  multiply  the  divisor  4,  by  this 

quotient  figure  3,  and  place  the  -product  12, 

24  under  the  figures  divided.     We  subtract  tho 

product  from   the  figures  divided,  and  have 

a  remainder  of  3.  Third.  Bringing  down  the  next  figure  of 
the  dividend  to  the  right  hand  of  the  remainder,  we  have  30, 
the  second  partial  dividend.  Then  4  is  contained  in  30,  7  times 
and  a  remainder.  Placing  the  7  at  the  right  hand  of  the  last 
quotient  figure,  and  multiplying  the  divisor  by  it,  we  place  the 
product  28,  under  the  figures  last  divided,  and  subtract  as 
before.  To  the  remainder  2,  bring  down  the  next  figure  4  of 
the  given  dividend,  and  we  have  24  for  the  third  partial  divi- 
dend. Then  4  is  contained  in  24,  6  times.  Multiplying  and 
subtracting  as  before,  we  find  that  nothing  remains,  and  we 
have  for  the  entire  quotient  376. 

NOTE.    When  the  whole  process  of  division  is  written  out  as  above,  the  operation 
is  termed  Long  Division.    The  principle  however  is  the  name  as  Short  Division. 

Solve  the  following  examples,  by  Long  Division. 

2.  Divide  4672  by  8.  .      Ans.       584. 

3.  Divide  97636  by  7.  Ans.  13948. 

4.  Divide  37863  by  9.  Ans.     4207. 
5    Divide  394064  by  11.  Am.  35824. 


DIVISION.  63 

6.  How  many  times  is  23  contained  in  17158  ? 

OPERATION.  ANALYSTS.     As  28  is  not  contained  in  the 

23)17158(746  first  two  figures  of  the  dividend,  we  find  how 

IQl  many  times  it  is  contained  in  171,  as  the  first 

partial  dividend*     23  is  contained  in  171,  7 

105  times,  which  we  place  in  the  quotient  on  the 

92  right  of  the  dividend.     We  then   multiply 

the  divisor  23,  by  the  quotient  figure  7,  and 

138  subtract  the  product  161,  from- the  part  of 

138  the  dividend  used,  and  we  have  a  remainder 

of  10.    To,  this  remainder  we  bring  down  the 

next  figure  of  the  dividend,  making  105  for  the  second  partial 
dividend.     Then,  23  is  contained  in  105,  4  times,  which  we 
place  in  the  quotient.     Multiplying  and  subtracting  as  before, 
we  have  a  remainder  of  13,  to  which  we  bring  down  the  next 
figure  of  the  dividend,  making  138  for  the  third  partial  divi- 
dend.    23  is  contained  in  138,  6  times;  multiplying  and  sub- 
tracting as  before,  nothing  remains,  and  we  have  for  the  entire 
quotient,  746. 

From  the  preceding  illustrations  we  derive  the  following 
general 

RULE.  I.  Write  the  divisor  at  the  left  of  the  dividend, 
as  in  short  division. 

II.  Divide  the  least  number  of  the  left  hand  figures  in 
the  dividend  that  will  contain  the  divisor  one  or  more  times, 
and  place  the  quotient  at  the  right  of  the  dividend,  with  a 
line  between  them. 

III.  Multiply  the  divisor  by  this  quotient  figure,  subtract 
the  product  from  the  partial  dividend  used,  and  to  the  re- 
mainder bring  down  the  next  figure  of  the  dividend. 

IV.  Divide  as  before,  until  all  the  figures  of  the  dividend 
have  been  brought  down  and  divided. 

V.  If  any  partial  dividend  will  not  contain  the  divisor, 


64  SIMPLE  NUMBERS. 

place  a  cipher  in  the  quotient,  and  bring  down  the  next 
figure  of  the  dividend,  and  divide  as  before. 

VI.  If  there  be  a  remainder  after  dividing  all  the  figures 
of  the  dividend,  it  must  be  written  in  the  quotient,  with  the 
divisor  underneath. 

NOTES.    1.  If  any  remainder  be  equal  to,  or  greater  than  the  divisor,  the  quotient 
figure  is  too  small,  and  must  be  increased, 

2.  If  the  product  of  the  divisor  by  the  quotient  figure  be  greater  than  the  partial 
dividend,  the  quotient  figure  is  too  large,  and  must  be  diminished. 

PROOF.     The  same  as  in  short  division. 

7 1 .  The  operations  in  long  division  consist  of  five  prin- 
cipal steps,  viz. : — 

1st.    Write  down  the  numbers. 

2d.     Find  how  many  times. 

3d.    Multiply. 

4th.  Subtract. 

5th.  Bring  down  another  figure. 


EXAMPLES   FOR  PRACTICE. 

7.  Find  how  many  times  18  is  contained  in  36838. 

OPERATION.  PROOF. 

Dividend.    Quotient. 

DiTisor,    18)36S38(2046jg  2046    Quotient. 

36  18    Divisor. 


83  16368 

72  2046 

118  36828 

108  10    Remainder. 

10  Remainder  36838    Dividend. 


DIVISION.  65 

8.  Divide  79638  by  36.  9.  Divide  93975  by  84. 

OPERATION.  OPERATION. 

86)79638(2212/j  84)93975(1118|j 

72  84 

76  99 

72  84 

43  157 

36  84 

~78  735 

72  672 

6    Rem.  63    Bern. 

10.  Divide  408722  by  136.  11.  Divide  104762  by  109. 

OPERATION.  OPERATION. 

136)408722(3005  109)104762(961 

408  981 


722 

680  654 


42  &».  122 

109 


12.  Divide  178464  by.16.  Am.  11154. 

13.  Divide  15341  by  29.  Ans.  529. 

14.  Divide  463554  by  39.  Ans.  11886. 

15.  Divide  1299123  by  17.  Ans.  76419. 

16.  Divide  161700  by  15.  An*.  10780. 

17.  Divide  47653  by  24. 

18.  Divide  765431  by  42. 


SIMPLE   NUMBERS. 


19.  Divide  6783  by  15. 

20.  Divide  7831  by  18. 

21.  Divide  9767  by  22. 

22.  Divide  7654  by  24. 

23.  Divide  767500  by  23. 

24.  Divide  250765  by  34. 

25.  Divide  5571489  by  43. 

26.  Divide  153598  by  29. 

27.  Divide  301147  by  63. 

28.  Divide  40231  by  75. 

29.  Divide  52761878  by  126. 

30.  Divide  92550  by  25. 

31.  Divide  7461300  by  95. 

32.  Divide  1193288  by  45. 

33.  Divide  5973467 -by  243. 

34.  Divide  69372168  by  342. 

35.  Divide  863256  by  736. 

36.  Divide  1893312  by  912. 

37.  Divide  833382  by  207. 

38.  Divide  52847241  by  607. 

39.  Divide  13699840  by  342. 

40.  Divide  946656  by  1038. 

41.  Divide  46447786  by  1234. 

42.  Divide  28101418481 

by  1107. 

43.  Divide  48288058.  by  3094. 

44.  Divide  47254149  by  4674. 

45.  A  man  bought  114  acres  of  land  for  4104  dollars , 
what  was  the  average  price  per  acre  ?       Ans.  36  dollars. 

46.  Nine  thousand  dollars  was  paid  to  75  operatives: 
how  much  did  each  receive?  Ans.  120  dollars. 


Quotients, 

Rem, 

452 

3. 

435 

1. 

443 

21. 

318 

22. 

33369 

13. 

7375 

15. 

129669 

22. 

5296 

14. 

4780 

7. 

536 

31. 

418745 

8.' 

3702 

78540 

26517 

23. 

24582 

41. 

202842 

204. 

1172 

664. 

2076 

4026 

87063 

40058 

4. 

912 

37640 

26. 

25385201 
15607 
10110 


974. 


9. 


DIVISION.  67 

47.  There  are  24  hours  in  a  day ;  how  many  days  in 
11424  hours  ?  Ans.  476. 

48.  In  one  hogshead  are  63  gallons ;  how  many  hogs- 
heads in  6615  gallons  ?  Ans.  105. 

49.  If  a  man  travel  48  miles  a  day,  how  long  will  it  take 
him  to  travel  1296  miles'?  Ans.  27  days. 

50.  If  a  person  can  count  8677  in  an  hour,  how  long 
will  it  take  him  to  count  38369694  ?     Ans.  4422  hours. 

51.  If  it  cost  5987520  dollars  to  construct  a  railroad  576 
miles  long,  what  will  be  the  average  cost  per  mile  ? 

Ans.  10395  dollars. 

52.  The  Memphis  and  Charleston  railroad  is  287  miles 
in  length,  and  cost  5572470  dollars;  what  was  the  average 
cost  per  mile  ?  Ans.  19416rr7587  dollars. 

53.  A  garrison  consumed  1712  barrels  of  flour  in  107 
days  ;  how  much  was  that  per  day  ?        Ans.  16  barrels. 

54.  How  long  would  it  take  a  vessel  to  sail  from  New 
York  to  China,  allowing  the  distance  to  be  9072  miles,  and 
the  ship  to  sail  144  miles  a  day  1  Ans.  63  days. 

55.  How  long  could  27  men  subsist  on  a  stock  of  provis- 
ion, that  would  last  1  man  3456  days  ?      Ans.  128  days. 

56.  A  drover  received  10362  dollars,  for  314  head  of  cat- 
tle ;  how  much  was  their  average  value  per  head  1 

Ans.  33  dollars. 

57.  If  42864  pounds  of  cotton  be  packed  in  94Jmles,  what 
is  the  average  weight  of  each  bale  1       Ans.  456  pounds. 

58.  If  a  field  containing  42  acres  produce  1659  bushels 
of  wheat,  what  will  be  the  numbor  of  bushels  per  acre  ? 

Ans.  39f  4  bushels. 

59.  In  what  time  will  a  reservoir  containing  109440  gal- 
lons, be  emptied  by  a  pump  discharging  608  gallons  per 
hour  1  Ans.  180  hours. 


68  SIMPLE    NUMBERS. 

CONTRACTIONS. 
CASE   I. 

72.  When  the  divisor  is  10,  100,  1000,  &c. 

1.  Divide  374  by  10. 

OPERATION.  ANALYSIS.     Since  we  have  shown, 

1!0^37'4  th&t  to  remove  a  figure  one  place  to- 

ward the  left  by  annexing  a  cipher 

Quotient,    37---4Rem.    increases  its  value  tenfold,  or  multi- 

or,  37T4Q,  Ans.      p\[cs  it  by  10,  so,  on  the  contrary,  by 

cutting  ofi   or  taking  away  the  right 

hand  figure  of  a  number,  each  of  the  other  figures  is  removed 
one  place  toward  the  right,  and,  consequently,  the  value  of  each 
is  diminished  tenfold,  or  divided  by  10. 

For  similar  reasons,  if  we  cut  off  two  figures,  we  divide  by 
100,  if  three,  we  divide  by  1000,  and  so  on.  Hence  the 

RULE.  From  the  right  hand  of  the  dividend  cut  of  as 
many  figures  as  there  are  ciphers  in  the  divisor.  Under  the 
figures  so  cut  off,  place  the  divisor,  and  the  Mchole  will  form 
the  quotient. 

EXAMPLES   FOR   PRACTICE. 

Quotients.          Kern's. 

2.  Divide  13705  by  100.  137  5. 

3.  Divide  50670  by  100.  506  70. 

4.  Divide  320762  by  1000.  320         762. 

5.  Divi<Jp  14030731  by  10000.  1403         731. 

6.  Divide  9021300640  by  100000.      90213         640. 

7.  A  man  sold  100  acres  of  land  for  3725  dollars ;  how 
much  did  he  receive  an  acre  ?  Ans.  37f\fty  dollars. 

8.  Bought  1000  barrels  of  flour  for  6080  dollars ;  how 
much  did  it  cost  me  a  barrel  ?  Ans.  6TH$jj  dollars. 

9.  Paid  12560  dollars  for  10000  bushels  of  wheat;  how 
much  was  the  cost  per  bushel  ?         Ans.  1^^^  dollars. 


DIVISION.  69 

CASE  III. 

73.  When  there  are  ciphers  on  the  right  hand 
of  the  divisor. 

1.  Divide  437661  by  800. 

OPERATION.  ANALYSIS.     In   this  example   we 

,     8'|00)4376i61  resojve  800  into  the  factors  8  and 

~547       61  Rem     ^°»  an(*  Divide  ^rst  ky  100,  by  cut- 

ting off  two  right  hand  figures  of  the 

dividend,  and  we  have  a  quotient  of  4376,  and  a  remainder  of  61. 
We  next  divide  by  8,  and  obtain  547  for  a  quotient  ;  and  the 
entire  quotient 


2.  Divide  34716  by  900. 

OPERATION.  ANALYSIS.    Dividing  as  in  the  last 

9jOO)347jl6  example,  we  have  a  quotient  of  38, 

Quotient,  38--  516  Rem.    and  a  remainder  of  5  after  dividing 

or    38  3  l&  Ans.      by  9'  which  we  prefix  to  the  fiSures 
cut  off  from  the  dividend,  making  a 

true  remainder  of  516,  and  the  entire  quotient  38|^.     Hence 
RULE.     I.   Cut  off  the  ciphers  from  the  right  of  the  divi- 

sor, and  the  same  number  of  figures  from  the  right  of  the 

dividend. 

II.  Divide  the  remaining  figures  of  the  dividend  by  the 

remaining  figures  of  the  divisior,  and  the  result  will  be  the 

quotient.     If  there  be  a  remainder  after  this  division,  pre- 

fix it  to  the  figures  cut  off  from  the  dividend,  and  this  will 

form  the  true  remainder. 

EXAMPLES   FOR   PRACTICE. 

Quotients.          Bern's. 

3.  Divide  46820  by  400.  117  20. 

4.  Divide  130627  by  800.  163  227. 

5.  Divide  76173  by  320.  238  13. 
6  Divide  378000  by  1200.  315 


70  SIMPLE  NUMBERS. 

7.  Divide  674321  by  11200.  60        2321. 

8.  Divide  64613214  by  4000.  16153         1214. 

9.  Divide  146200  by  430.  340 

10.  Divide  7380964  by  23000.  320       20964. 

11.  Divide  58677000  by  1800.          32598  600. 

EXAMPLES    IN    THE   PRECEDING   RULES. 

1.  A  speculator  bought  at  different  times  320  acres,  175 
acres,  87  acres,  and  32  acres  of  land,  and  afterward  sold  467 
acres ;  how  many  acres  had  he  left  1         Ans.  147  acres. 

2.  Two  men  travel  in  opposite  directions ;  one  travels  31 
miles  a  day,  the  other  43  miles  a  day ;  how  far  apart  will 
they  be  in  12  days  ?  Ans.  888  miles. 

3.  A  tobacconist  has  6324  pounds  of  tobacco,  which  he 
wishes  to  pack  in  boxes  containing  62  pounds  each ;  bow 
many  boxes  must  he  procure  to  contain  it  ?        Ans.  102. 

4.  A  farmer  sold  15  tons  of  hay  at  9  dollars  a  ton,  and 
25  cords  of  wood  at  4  dollars  a  cord,  and  wished  to  divide 
the  amount  equally  among  5  creditors ;  how  much  would 
each  receive  ?  Ans.  47  dollars. 

5.  If  you  deposit  216  cents  each  week  in  a  savings  bank, 
and  take  out  89  cents  a  week,  how  many  cents  will  you 
have  in  bank  at  the  end  of  36  weeks  ?     Ans.  4572  cents. 

6.  The  product  of  two  numbers  is  8928,  and  one  of  the 
numbers  is  72  ;  what  is  the  other  number  ?       Ans.  124. 

7.  The  dividend  is  7280,  and  the  quotient  is  208 ;  what  is 
the  divisor  1  Ans.  35. 

8.  What  is  the   remainder    after    dividing  876437  by 
16900  1  Ans.  14537. 

9.  A  man  sold  6  horses  at  125  dollars  each,  25  head  of 
cattle  at  30  dollars  each,  and  with  the  proceeds  bought 
land  at  25  dollars  an  acre  ;  how  many  acres  did  he  buy  1 

Ans.  60  acres. 


PROMISCUOUS  EXAMPLES.  71 

10.  If  a  Mechanic  receives  784  dollars  a  year  for  labor, 
and  his  expenses  are  426  dollars  a  year,  how  much  can  he 
gaspe  in  6  years  ?  Ans.  2148  dollars. 

.  11.  A  farmer  sold  40  bushels  of  wheat  at  2  dollars  a 
bushel,  and  16  cords  of  wood  at  3  dollars  a  cord.  He  re- 
ceived 15  yards  of  cloth  at  4  dollars  a  yard,  and  the  re- 
mainder in  money ;  how  much  money  did  he  receive  ? 

Ans.  68  dollars. 

12.  How  many  pounds  of  cheese  worth  10  cents  a  pound, 
can  be  bought  for  22  pounds  of  butter  worth  15  cents  a 
pound  ?  Ans.  33  pounds. 

13.  If  56  yards  of  cloth  cost  336  dollars,  how  much  will 
12  yards  cost,  at  the  same  rate  ?  Ans.  72  dollars. 

14.  If  100  barrels  of  flour  cost  600  dollars,  what  will 
350  barrels  cost,  at  the  same  rate  ?       Ans.  2100  dollars. 

15.  How  long  can  60  men  subsist  on  an  amount  of  food 
that  will  last  1  man  7620  days  ?  Ans.  127  days. 

16.  If  I  buy  225  barrels  of  flour  for  1125  dollars,  and 
sell  the  same  for  1800  dollars,  how  much  do  I  gain  on  each 
barrel?  Ans.  8  dollars. 

17.  A  man  sold  his  house  and  lot  for  5670  dollars,  and 
took  his  pay  in  bank  stock  at  90  dollars  a  share ;  how  many 
shares  did  he  receive  ?  Ans.  68  shares. 

18.  How  many  pounds  of  tea  worth  75  cents  a  pound, 
ought  a  man  to  receive  in  exchange  for  27  bushels  of  oats, 
worth  50  cents  a  bushel  ?  Ans.  18  pounds. 

19.  The  quotient  of  one  number  divided  by  another  is 
40,  the  divisor  is  364,  and  the  remainder  120 ;  what  is  the 
dividend?  Ans.  14680. 

20.  How  many  tons  of  hay  at  12  dollars  a  ton,  must  be 
given  for  21  cows  at  24  dollars  apiece  ?       Ans.  42  tons. 


72  SIMPLE  NUMBEKS. 

21.  Bought  150  barrels  of  flour  for  1050  dollars,  and  sold 
107  barrels  of  it  at  9  dollars  a  barrel,  and  the  remainder  at 
7  dollars  a  barrel ;  did  I  gain  or  lose,  and  how  much  ? 

Ans.  gained  214  dollars. 

22.  A  mechanic  earns  45  dollars  a  month,  and  his  neces- 
sary expenses  are  27  dollars  a  month.     How  long  will  it 
take  him  to  pay  for  a  farm  of  50  acres,  at  27  dollars  an 
acre  ?  Ans.  75  months. 

23.  How  many  barrels  of  flour  at  7  dollars  a  barrel,  will 
pay  for  30  tons  of  coal,  at  4  dollars  a  ton,  and  44  cords  of 
wood,  at  3  dollars  a  cord  ?  Ans.  36  barrels. 

PROBLEMS   IN   SIMPLE  INTEGRAL   NUMBERS. 

T4.  The  four  operations  that  have  now  been  considered, 
viz.,  Addition,  Subtraction,  Multiplication,  and  Division,  are 
all  the  operations  that  can  be  performed  upon  numbers,  and 
hence  they  are  called  the  Fundamental  Rules* 

Tt>.  In  all  cases,  the  numbers  operated  upon  and  the  re- 
sults obtained,  sustain  to  each  other  the  relation  of  a  whole 
to  its  parts.  Thus, 

I.  In  Addition,  the  numbers  added  are  the  parts,  and  the 
sum  or  amount  is  the  whole. 

II.  In  Subtraction,  the  subtrahend  and  remainder  are  the 
parts,  and  the  minuend  is  the  whole. 

III.  In  Multiplication,  the  multiplicand  denotes  the  val- 
ue of  one  part,  the  multiplier  the  number  of  parts,  and  the 
product  the  total  value  of  the  whole  number  of  parts. 

IV.  In  Division,  the  dividend  denotes  the  total  value  of 
the  whole  number  of  parts,  the  divisor  the  value  of  one  part, 
and  the  quotient  the  number  of  parts ;  or  the  divisor  the 
number  of  parts,  and  the  quotient  the'  value  of  one  part. 


PROBLEMS.  3 

76.  Let  the  pupil  be  required  to  illustrate  the  following 
problems  by  original  examples. 

Problem  1.  Given,  several  numbers,  to  find  their  sum. 

Prob.  2.  Given,  the  sum  of  several  numbers  and  all  of 
them  but  one,  to  find  that  one. 

Pi*ob.  3.  Given,  two  numbers,  to  find  their  difference. 

Prob.  4.  Given,  the  minuend  and  subtrahend,  to  find  the 
remainder. 

Prob.  5.  Given,  the  minuend  and  remainder,  to  find  tha 
subtrahend. 

Prob.  6.  Given,  the  subtrahend  and  remainder,  to  find 
the  minuend. 

Prob.  7.  Given,  two  or  more  numbers,  to  find  their  prod- 
uct. 

Prob.  8.  Given,  the  multiplicand  and  multiplier,  to  find 
the  product. 

Prob.  9.  Given,  the  product  and  multiplicand,  to  find  the 
multiplier. 

Prob.  10.  Given,  the  product  and  multiplier,  to  find  the 
multiplicand. 

Prob.  11.  Given,  two  numbers,  to  find  their  quotitots. 

Prob.  12.  Given,  the  divisor  and  dividend,  to  find  the 
quotient. 

Prob.  13.  Given,  the  divisor  and  quotient,  to  find  the 
dividend. 

Prob.  14.  Given,  the  dividend  and  quotient,  to  find  the 
divisor. 

Prob.  15.  Given,  the  divisor,  quotient,  and  remainder,  to 
find  the  dividend. 

Prob.  16.  Given,  the  dividend,  quotient,  and  remainder 
to  find  the  divisor. 


74  FRACTIONS. 


FRACTIONS. 

DEFINITIONS,   NOTATION,   AND   NUMERATION. 

7  7.  If  a  unit  be  divided  into  2  equal  parts,  one  of  the 
parts  is  called  one  half. 

If  a  unit  be  divided  into  3  equal  parts,  one  of  the  parts  is 
called  one  third,  two  of  the  parts  two  thirds. 

If  a  unit  be  divided  into  4  equal  parts,  one  of  the  parts  is 
called  one  fourth,  two  of  the  parts  two  fourths,  three  of  the 
parts  three  fourths. 

If  a  unit  be  divided  into  5  equal  parts,  one  of  the  parts  is 
called  one  fifth,  two  of  the  parts  two  fifths,  three  of  the 
parts  three  fifths,  &c. 

And  since  one  half,  one  third,  one  fourth,  and  all  other 
equal  parts  of  an  integer  or  whole,  thing,  are  each  in  them- 
selves entire  and  complete,  the  parts  of  a  unit  thus  used 
are  called  fractional  units  ;  and  the  numbers  formed  from 
them,  fractional  numbers.  Hence 

7£l£  A  Fractional  Unit  is  one  of  the  equal  parts  of  an 
integral  unit. 

TO.  A  Fraction  is  a  fractional  unit,  or  a  collection  of 
fractional  units. 

8O.  Fractional  units  take  their  name,  and  their  value, 
from  the  number  of  parts  into  which  the  integral  unit  is 
divided.  Thus,  if  we  divide  an  orange  into  2  equal  parts, 
the  parts  are  called  halves;  if  in  to  3  equal  parts,  thirds; 
if  into  4  equal  parts,  fourths,  &c. ;  and  each  third  is  less  in 
value  than  each  half,  and  e&c\i  fourth  less  than  each  third} 
and  the  greater  the  number  of  parts,  the  leas  their  value. 


DEFINITIONS,  NOTATION,  AND  NUMERATION.      75 
The  parts  of  a  fraction  are  expressed  by  figures ;  thus, 


One  half  is  written  A 

One  third         "  ^ 

Two  thirds       "  f 

One  fourth        "  \ 

Two  fourths      «  f 

Three  fourths   «  f 


" 


One  fifth  is  written 
Two  fifths 
One  seventh  "  ^ 
Three  eighths  "  f 
Five  ninths  "  f 

Eiht  tenths         "         - 


To  write  a  fraction,  therefore,  two  integers  are  required, 
one  written  above  the  other  with  a  line  between  them. 

8  1 .  The  Denominator  of  a  fraction  is  the  number  below 
the  line.  It  shows  into  how  many  parts  the  integer  or  unit 
is  divided,  and  determines  the  value  of  the  fractional  unit. 

82.  The  Numerator  is  the  number  above  the  line.     It 
numbers  the  fractional  units,  and  shows  how  many  are 
taken. 

83.  Thus,  if  one  dollar  be  divided  into  4  equal  parts, 
the  parts  are  called  fourths,  the  fractional  unit  being  one 
fourth,  and  three  of  these  parts  are  called  three  fourths  of  a 
dollar,  and  may  be  written 

3  the  number  of  parts  or  fractional  units  taken. 

4  the  number  of  parts  or  fractional  units  into  which  the  dollar  is  divided. 

84.  The  Terms  of  a  fraction  are  the  numerator  and 
denominator,  taken  together. 

80.  Fractions  indicate  division,  the  numerator  answer- 
ing to  the  dividend,  and  the  denominator  to  the  divisor. 
Hence, 

86.  The  Value  of  a  fraction  is  the  quotient  of  the  nu- 
merator divided  by  the  denominator. 

Thus ;  the  quotient  of  4  divided  by  5  is  |,  or  J  expresses 

<he  QUOtie/lt  of  Which  {  1  is  the  dividend. 
•  \  5  is  the  divisor. 


76  FK  ACTIONS. 

1.  What  is  1  half  of  8? 

ANALYSIS.  It  is  the  quotient  of  8  divided  by  2,  which  is  4; 
or,  it  is  a  number,  which  taken  2  times,  will  make  8,  which  is  4. 
Therefore,  4  is  1  half  of  8. 

2.  What  is  2  thirds  of  9  ? 

ANALYSIS.  Since  1  third  of  9  is  3,  2  thirds  of  9  is  2  times  3, 
which  is  6.  Therefore,  2  thirds  of  9  is  6. 

Hence,  to  obtain  one  half,  one  third,  one  fourth,  or  any 
fractional  part  of  a  number,  we  divide  that  number  by  the 
denominator  of  the  fraction  expressing  the  parts  ;  and  to 
obtain  any  given  number  of  such  parts,  we  multiply  that 
part  by  the  number  of  parts  expressed  by  the  numerator 
of  the  same  fraction. 

8.  What  is  1  fourth  of  12  ?  3  fourths  of  12  ? 

4.  What  is  1  fifth  of  20  ?  3  fifths  ?  4  fifths  ? 

5.  What  is  1  eight  of  40  ?    3  eighths  ?  5  eighths  ? 

6.  What  is  2  sevenths  of  2^1  ?  5  sevenths  of  35  ?  6  sev- 
enths of  49? 

7.  What  is  1  ninth  of  63  ?  2  ninths  of  27  ?  4  ninths  of 
36  ?  5  ninths  of  45  ?  7  ninths  of  81  ? 

8.  What  is  1  twelfth  of  48  ?  5  twelfths  ?  7  twelfths  ? 

9.  If  a  pound  of  coffee  cost  15  cents,  how  much  will  1 
third  of  a  pound  cost  ?  2  thirds  ? 

10.  A  farmer  having  60  sheep,  sold  1  fifth  of  them  to 
one  man,  and  3  fifths  to  another ;  how  many  did  he  sell  to 
both? 

11.  A  boy  having  48  cents,  spent  5  eighths  of  them ; 
how  many  had  he  left  ? 

12.  Paid  108  dollars  for  a  horse,  and  9  twelfths  as  much 
for  a  carriage ;  how  much  did  the  carriage  cost  ? 

13.  William  had  120  pennies,  and  James  had  7  tenths 
as  many  ;  how  many  had  James  ? 


NOTATION   AND   NUMERATION.  77 

87.  It  is  often  required  to  express  by  a  fraction,  what 
part  one  number  is  of  another  number. 

1.  What  part  of  5  is  3  ? 

ANALYSIS.     Since  1  is  1  fifth  of  5,  3  must  be  3  times  1  fifth  of 
5,  or  3  fifths  of  5.     Therefore,  8  is  8  fifths  of  5. 

NOTK.    The  number  preceded  by  the  word  of  is  generally  made  the  denominator 
or  divisor,  and  the  other  number  called  the  part,  the  numerator  or  dividend. 

2.  What  part  of  6  is  3?  4?  5?  1? 

3.  What  part  of  9  is  2?  3?  5?  6?  1?  4? 

4.  What  part  of  10  is  7?  6?  3?  1?  9?  8?  4? 

6.  What  part  ot  12  is  a?  5?  6?  8?  9?  11  10?  11? 

6.  What  part  of  14  is  5?  7?  9?  3?  6?  11?  8?  15? 

7.  What  part  of  15  bushels,  is  3  bushels  ?  7  bushels  ?  9 
bushels  11  bushels? 

8.  What  part  of  18  dollars,  is  7  dollars?  5  dollars?  9 
dollars  ?  17  dollars  ? 

9.  If  6  oranges  cost  30  cents,  what  part  of  30  cents  will 
1  orange  cost  ?  2  oranges  ?  3  oranges  ?  5  oranges  ? 

EXAMPLES   IN  WRITING  AND  READING  FRACTIONS. 
JSxpress  the  following  fractions  by  figures  : — 

1.  Nine  twelfths.  Ans.  T%. 

2.  Eleven  fifteenths.  Ans.  jj. 
8.  Twenty-four  for ty-ninths.  Ans.  ||. 

4.  Forty-four  sixty-ninths.  Ans.  ||. 

5.  One  hundred  twenty,  four  hundred  fiftieths. 

Read  the  the  following  fractions  : 

6-  ft,  ii,  ii,  flfc,  M,M,  W.  If?- 

7.  If  the  fractional  unit  is  28,  express  9  fractional  units, 
16,  17;  22;  27. 

8*  If  the  fractional  unit  is  96,  express  27  fractional 
units ;  42 ;  75 


78  FRACTIONS. 

88.  Fractions  are  distinguished  as  Proper  and  Improper. 
A  Proper  Fraction  is  one  whose  numerator  is  less  than 

its  denominator ;  its  value  is  less  than  the  unit,  1 .     Thus, 
T72>  75g>  T°o>  II  are  proper  fractions. 

An  Improper  Fraction  is  one  whose  numerator  equals 
or  exceeds  its  denominator ;  its  value  is  never  less  than  the 
unit,  1.  Thus,  ^,  |,  -Lo.?  jyi,  |o?  J^Q  are  improper  fractions. 

89.  A  Mixed  Number    is  a  number  expressed  by  an 
integer  and  a  fraction ;  thus,  4j,  17Jf ,  9T%  are  mixed  num- 
bers. 

REDUCTION. 

....~  9O.  The  Reduction  of  a  fraction  is  the  process  of  chang- 
ing its  terms,  or  its  form,  without  altering  its  value. 

CASE   I. 

91.  To  reduce  fractions  to  their  lowest  terms. 

A  fraction  is  in  its  lowest  terms  when  no  number  greater 
than  1  will  exactly  divide  both  numerator  and  denominator 
without  a  remainder. 

1.  Reduce  f  to  its  lowest  terms.  ^ 

ANALYSIS.     It  is  plain,  that  the  numerator  2,  and  the  denom- 
inator 4,  are  both  divisible  by  2,  without  remainders;  hence 
2-j-2_l 
4-f-2~2 

The  terms  thus  obtained,  viz.,  1,  the  numerator,  and  2,  the  de- 
nominator, are  not  divisible  by  any  number  larger  than  1,  and 
therefore  are  the  smallest  terms  by  which  the  value  of  £  can  be 
expressed. 

2.  Reduce   |  to  its  lowest  terms. 

3.  Reduce  -f^  to  its  lowest  terms.  _ 

4.  Reduce   |   to  its  lowest  fe rnis.  . '/. 

5.  Reduce  -£0  to  its  lowest  terms. 

6.  Reduce  '  jj  to  it*  lowest  forms. 


REDUCTION.  79 

7.  Reduce  ||  to  its  lowest  terms. 

OPERATION.  ANALYSIS.     Dividing  both  terms 

9V8  — 24  .  9Y24 — i_2  •  of  a  fraction  by  the  same  number 

^/gO  —  3l)>   TVSO 15' 

o\i2 4     A  does  not  alter  the  value  of  the  frac- 

J°      "g      4     ,  tion  or  quotient;  hence,  we  divide 

both  terms  of  j  £   by  2,  and  obtain 

|J;  dividing  both  terms  of  this  fraction  by  2,  we  have  ||  as 
the  result ;  finally,  dividing  the  terms  of  this  fraction  by  3,  we 
have  |,  and  as  no  number  greater  than  1  will  divide  the  terms 
of  this  fraction  without  a  remainder,  |  are  the  lowest  terms 
in  which  the  value  of  |J  can  be  expressed.  We  may  obtain 
the  final  result  more  readily,  by  dividing  the  terms  of  this  frac- 
tion by  the  largest  number  that  will  -divide  both  without  a  re- 
mainder, as  in  the  above  example  ;  if  we  divide  by  12,  we  obtain 
£,  the  answer.  Hence  the 

RULE.  Divide  the  terms  of  the  fraction  by  any  numbei 
greater  than  1,  that  will  divide  both  without  a  remainder, 
and  the  quotients  obtained  in  the  same  manner,  until  no  num- 
ber greater  than  1  will  so  divide  them  ;  the  last  quotients 
will  be  the  lowest  terms  of  the  given  fraction. 


it? 


EXAMPLES   FOR   PRACTICE. 

8.  Reduce   ^J  to  its  lowest  terms.  Ans.  |. 

9.  Reduce  y7^  to  its  lowest  terms.  Ans.  f . 
^0.  Reduce  T9T82  to  its  lowest  terms.  Ans.  g. 

11.  Deduce  jff  to  its  lowest  terms.  Ans.  J. 

12.  Reduce  7||  to  its  lowest  terms.  **Ans.  l|. 

13.  Reduce  |4|  to  its  lowest  terms.  Ans.  7|£ 

14.  Reduce  HI  to  its  lowest  terms.  Ans.  f, 

15.  Reduce  -ff^  to  its  lowest  terms.  Ans.  ^. 

16.  Reduce  ||g  to  its  lowest  terms.  ^4ns.  ||. 
• — -17.  Reduce  3^ff  to  its  lowest  terms.  Ans.  i. 

18.  Reduce  ||J  to  its  lowest  terms.  Ans.  ^%. 

19.  Reduce  $g-J-jj  to  its  lowest  terms.  ^4n*.  j^f  J. 


80  FRACTIONS. 

CASE  II. 

93.  To  change  an  improper  fraction  to  a  whole 
or  mixed  number. 

1.  In  if  how  many  times  1  ? 

ANALYSIS.  Since  1  equals  J,  L2  equal  as  many  times  1,  as 
|  are  contained  times  in  \f,  which  are  3  times.  Therefore, 
L2  are  3  times  1,  or  3. 

2.  How  many  times  1  in  y  ?  in  Jg8  ?  in  \°  ? 

3.  How  many  times  1  in  2^  ?  in  2g4  ?  in  3/  ? 

4.  How  many  times  1  in  fig4  ?  in  f  g  ?  in  4g8  ? 

5.  How  many  times  1  in  7^2  ?  in  ff  ?  in  f  f  ? 

Nora.  When  the  denominator  is  not  an  exact  divisor  of  the  numerator,  the  re- 
sult will  be  a  mixed  number. 

6.  In  \f  how  many  times  1  1 

OPERATION.  ANALYSIS.    Since  1  equals  ^,  \f  equal 

as  many  times  1  as  7  is  contained  times 
in  16,  which  is  2S  times.     Hence  the 


, 


2  1  Ans. 

Divide  the  numerator  by  the  denominator. 

EXAMPLES    FOR   PRACTICE. 


7.  In  J-ffi-  how  many  times  1  ?  u4ns.  244. 

8.  In  2T228  of  a  year  how  many  years  ?  Ans.  19. 

9.  In  12-|4  of  a  pound  how  many  pounds?    Ans.  107." 

10.  In  m  of  a  mile  how  many  miles  1  Ans.  6. 

11.  In  7JC7  of  a  rod  how  many  rods?        Ans.  21£J. 

12.  In  2f$5  of  a  dollar  how  many  dollars? 

13.  Reduce  f  g  to  a  whole  number.  Ans.  6. 

14.  Reduce  y/  to  a  mixed  number.  Ans.  5|. 

15.  Reduce  87254  to  a  whole  number.  -4ns.  18. 
Reduce  ^f6  to  a  mixed  number.    *        Ans.  60f. 

17.  Change  3||6  to  a  mixed  number.  67|. 

18.  Change  2j|4  to  a  whole  number.  An*.  52. 


REDUCTION.  81 

CASE  m. 

93.  To  reduce  a  whole  or  mixed  number  to  an  im- 
proper fraction. 

1.  How  many  thirds  in  4  ? 

ANALYSIS.  Since  in  1  there  are  3  thirds,  in  4  there  are  4 
times  3  thirds,  or  12  thirds.  Therefore,  there  are  1*  in  4. 

2.  How  many  fourths  in  2  1  in  3  ?  in  5  ? 

3.  How  many  halves  in  5  ?  in  7  ?  in  8  ?  in  9  ? 

4.  How  many  sixths  in  3  ?  in  5  ?  in  7  ?  in  10  ? 

5.  How  many  tenths  in  4  ?  in  8  1  in  9  ?  in  6  ? 

6.  How  many  fifths  in  2  whole  oranges  ?  in  4  1   in  5  ? 

7.  How  many  eighths  in  4  whole  dollars  ?  in  5  ?  in  6  ? 

8.  In  3|  dollars  how  many  eighths  of  a  dollar  ? 

OPERATION. 

35  ANALYSIS.     Since  in  1  dollar  there  are  8 

gH  eighths,   in    3    dollars    there    are    3    times  8 

eighths,  or  24  eighths,  and  5  eighths  added, 
24-£5  =  \9  make-2/-.' 

BULE.  Multiply  the  whole  number  by  the  denominate* 
of  the  fraction  ;  to  the  product  add  the  numerator,  and  un- 
der the  result  write  the  denominator. 

EXAMPLES   FOR   PRACTICE. 

9.  Reduce  6|  to  an  improper  fraction.  Ans.  2?7. 

10.  Reduce  7f  to  an  improper  fraction.  Ans.  6^8. 

11.  Reduce  15  to  a  fraction  whose  denominator  is  7. 

Ans.   ij}«. 

12.  Reduce  120  to  twelfths.  Ans.   Jf  J°. 

13.  In  242|  of  an  acre  how  many  thirds  of  an  acre  ? 

14.  In  75|  bushels  how  many  eighths  f         Ans.  6g7. 
^—  15.  In  24  pounds  how  many  sixteenths?       Ans.  sT8g4. 

16.  In  52  weeks  how  many  sevenths?  Ans.   3§4. 

17.  Change  14^  to  an  improper  fraction.     Ans.   \\6t 


62        '  ITKACTION8. 

CASE  IV. 

94.  T  3  reduce  two  or  more  fractious  to  a  com- 
mon denominator. 

A  Common  Denominator  is  a  denominator  common  to 
two  or  more  fractions. 

NOTK.  Any  number  that  can  be  divided  by  each  of  the  denominators  of  the 
given  fractions,  may  be  taken  for  the  common  denominator. 

1.  Reduce  \  an,d  f  to  fractions    having  a  common  de- 
nominator. 

ANALYSIS.  12  is  exactly  divisible  by  4  and  3,  and  may  there- 
fore be  taken  for  a  common  denominator.  Since  in  1  there  are 
12,  in  -1  of  1  there  must  be  1  of  If  or  J^  •  and  in  |  of  1  there 
must  be  |  of  ||,  or  ^.  Therefore  1  and  |  are  equal  to  ^ 

••**•  ' 

2.  Reduce  |  and  |  to  a  common  denominator.  * 

3.  Reduce  |  and  |  to  a  common  denominator. 

4.  Reduce  -J  and  |  to  a  common  denominator.'  . 

5.  Reduce  J-  and  |  to  a  common  denominator!  * 

OPERATION.  ANALYSIS.     We  multiply  the  terms  of  the 

__25          first  fraction  |,  by  the  denominator  5  of  the 
—SQ          second,  and  the  terms  of  the  fraction  |,  by 
the  denominator  6  of  the  first.     This  must  re- 
_          duce  each  fraction  to  the  same  denominator 


gQ^  for  gg^jj  new  denommator  will  be  the  pro 
duct  of  the  given  denominators.     Hence  the 
RULE.  Multiply  both  terms  of  each  fraction  by  the  d&» 
nominators  of  all  the  other  fractions. 

NOIK.    Mixed  numbers  must  first  be  reduced  to  improper  fractions.  . 
EXAMPLES   FOR   PRACTICE. 

6.  Reduce  %  and  |  to  a  common  denominator. 

An,.  - 


ADDITION.  88 

7.  .Reduce  j  and  |   to  a  common  denominator. 

/«.  H,  U- 

8.  Reduce  |  and  |   to  a  common  denominator. 

AM.  jf,  Jf. 

9.  Reduce  |  and  T7^   to  a  common  denominator. 

Ans.  J  j,  I?. 

10.  Reduce  |  and  T52   to  a  common  denominator. 

Am.  if,  ti- 

11. Reduce  £,  f  ,  and  £  to  a  common  denominator. 

Am.  if,  jj,  if. 

12.  Reduce  j,  |,  and  ^   to  a  common  denominator. 


13.  Reduce  |,  J,  and  |  to  a  common  denominator. 


14.  Reduce  1^,  f  ,  and  |  to  a  common  denominator. 

Ans.  J^8-,  f4,  f 

15.  Reduce  T7^,  2|,  and  f  to  a  common  denominator. 


16.  Reduce  -f^,  3^,  |,  and  |  to  a  common  denominator. 


I,  , 


ADDITION. 

95.  The  denominator  of  a  fraction  determines  the  value 
of  the  fractional  unit;  hence, 

I.  If  two  or  more  fractions  have  the  same  denominator, 
their  numerators  express  fractional  units  of  the  same  value. 

II.  If  two  or  more  fractions  have  different  denominators, 
their  numerators  express  fractional  units  of  different  values. 

And  since  units  of  the  same  value  only  can  be  united  into 
one  sum,  it  follows, 

III.  »That  fractions  can  be  added  only  when  they  have 
the  same  fractional  unit  or  common  denominator. 


84  FRACTIONS. 

1.  What  is  the  sum  of  i,  1,1,1? 

ANALYSIS.  When  fractions  have  a  coinmor  denominator, 
their  sum  is  found  by  adding  their  numerators,  and  placing  the 
sum  over  the  common  denominator.  Thus,  1+34-4  +  2=10, 
the  sum  of  the  numerators ;  placing  this  sum  over  the  common 
denominator  5,  we  have  L°— 2,  the  required  sum. 

2.  What  is  the  sum  of  T30,  T4y  and  T^  ? 

3.  What  is  the  sum  of  f ,  f ,  4  and  «  ? 

4.  What  is  the  sum  of  J,  f ,  £,  f  and  f  ? 

5.  A  boy  paid  |  of  a  dollar  for  a  pair  of  gloves,  §  of  a 
dollar  for  a  knife,  and  J  of  a  dollar  for  a  slate ;  how  much 
did  he  pay  for  all  1 

6.  A  father  distributed  some  money  among  his  children, 
as  follows  :  to  the  first  he  gave  -f^  of  a  dollar,  to  the  second 
T32,  to  the  third  T73,  to  the  fourth  T92,  and  to  the  fifth  T42  ; 
how  much  did  he  give  to  all  ? 

7.  What  is  the  sum  of  f  and  f  ? 

OPERATION.  ANALYSIS.  As  the  giv- 

|-f  |=||+ 58ff=f  |  Ans.  en  fractions  have  not  a 
common  denominator,  we  reduce  them  to  the  same  fractional 
unit,  (94)  and  then  add  their  numerators,  27+8—35,'  placing 
the  sum  over  the  common  denominator  36,  we  obtain  ||- 
hence  the  following 

RULE.  I.  When  the  given  fractions  have  the  same  de- 
nominator, add  the  numerators,  and  under  the  sum  write  the 
common  denominator. 

II.  When  they  have  not  the  same  denominator,  reduce 
them  to  a  common  denominator,  and  then  add  as  before. 

NOTE.  If  the  amount  be  an  improper  fraction,  reduce  it  to  A  whole  or  a  mixed 
number. 

EXAMPLES  FOR  PRACTICE. 

8.  What  is  the  sum  of  f  and  |  ?  Ans.  1T75. 

9.  What  is  the  sum  of  J  and  f  ?  Ans.  \\. 


2  /  ADDITION.  85 

10.  What  is  the  sum  of  f  and  §  ?  Ans.  |J. 

11.  Add  |,  |  and  |  together.  Ans.  1J. 

12.  Add  -f ,  £  and  f  together.  Ans.  l-J^. 

13.  Add  T%,  |,  |  and  \  together.  Ans.  2f . 

14.  Add  3,  £jtnd  |^  togetlier. 

15.  Add  |/4,  |  and  f  together. 

16.  What  is  the  sum  of  |,  f  and  «  1  Ans. 

17.  What  is  the  sum  of  f ,  |  and  |  ?  ^4ras.  If  1. 

18.  What  is  the  sum  of  f ,  f  and  1 1  Ans.  2  fSL9 
To  add  mixed  numbers,  add  the  fractions  and  integers 

separately,  and  then  add  their  sums. 

•  NOTE.    If  the  mixed  numbers  are  small,  they  may  be  reduced  to  improper 
fractions,  and  then  added  after  the  usual  method. 

19.  What  is  the  sum  of  14|,  21£  and  9|  ? 

OPERATION.  ANALYSIS.     By  reducing  the  frac- 

141  =  14^  tions  to  a  common  denominator,  and 

214 =21A$  adding  them,  we  obtain  ||  or  1^.£, 

93=  944  which  added  to  the  sum  of  the  inte- 

45P  Ans.       £ral  numbers>  Sives  45il' the  Ans. 

20.  What  is  the  sum  of  3|,  12|  and  25f  ?    Ans.  41|. 

21.  What  is  the  sum  of  |,  15£,  42-J  and  50  ? 

22.  What  is  the  sum  of  30|,  1J,  16^  and  ||? 

23.  Bought  3  pieces  of  cloth  containing  45^,  881,  and 
35|  yards ;  how  many  yards  in  the  3  pieces  ? 

Ans.  119^2  yards. 

24.  Three  men  bought  a  horse.     A  paid  31|  dollars,  B 
paid  43T53  dollars,  and  C  paid  47 1  dollars ;  what  was  the 
cost  of  the  horse  1  Ans.  122§  dollars. 

25.  If  it  take  5^  yards  of  cloth  for  an  overcoat,  4|  yards 
for  a  dress  coat,  2|  yards  for  a  pair  of  pantaloons,  and  |  of 
a  yard  for  a  vest,  how  many  yards  of  cloth  will  it  take  for 
the  whole  suit?  Ans.  12|  yards. 


. 

• 

86  FRACTIONS.  ^   * 

SUBTRACTION. 

96.  The  process  of  subtracting  one  fi  action  from  anoth- 
er is  based  upon  the  following  principles  : 

I.  One  number    can  be  subtracted  from  another   only 
when  the  two  numbers  have  the  same  unit  valifc.     Hence, 

II.  ?h  subtraction  of  fractions,  the  minuend  and  subtra- 
hend must  have  a  common  denominator, 

1.  From  T92  subtract  T63. 

ANALYSIS.  Since  the  fractions  have  a  common  denominator, 
the  difference  is  obtained,  by  subtracting  the  less  numerator  5, 
from  the  greater  9,  and  writing  the  result  over  the  common  der 
nominator  12 ;  we  thus  obtain  J^  the  required  difference. 

2.  From  |  subtract  f . 

3.  From  jj  subtract  T\. 

4.  Subtract  4J  from  f|. 

5.  James  had  J  of  a  bushel  of  walnuts,  and  sold  |  of 
them  ]  how  many  had  he  left  ? 

6.  Harvey  had  jf  of  a  dollar,  and  gave  T5ff  of  a  dollar  to 
a  beggar ;  how  much  had  he  left  ? 

7.  Subtract  |  from  f . 

OPERATION.  ANALYSIS.   As  the  given  frac- 

|  —  f  =  2i — izj^/r  <4**f.  tions  have  not  a  common  de- 
nominator, we  first  reduce  them  to  the  same  fractional  unit, 
(94)  and  then  subtract  the  less  numerator  9,  from  the  greater 
14,  and  write  the  result  over  the  common  denominator  21.  We 
thus  obtain  55T  the  required  difference.  Hence  the  following 

RULE.  I.  When  the  fractions  have  the  same  denomina- 
tor, subtract  the  less  numerator  from  the  greater,  and  place 
the  result  over  the  common  denominator. 

II.  When  they  have  not  a  common  denominator,  reduce 
tliem  to  a  common  denominator  before  subtracting. 

* 

t  K    %    V 


SUBTRACTION".  87 

EXAMPLES   FOR   PRACTICE. 

8.  From  J  take  f .  Ans.  j. 

9.  From  |  take  f  Ans.  £. 

10.  From  f  take  f.  ,4ns.  -JJ. 

11.  From  Jjj  take  £.  ^TW.  -H- 

12.  Subtract  f  from  f .  ^ns.  /,. 

13.  Subtract  ^  from  f  Ans.  fc 

14.  Subtract  f  from  ij. 

15.  Subtract-^  from  jj. 

16.  Subtract  ^  from  |.  <4ns.  2\. 

17.  Subtract  |  J  from  J.  ^Ins.  ^. 

18.  From  9|  take  2|. 

OPERATION.  ANALYSTS.     We  first  reduce  the  frac- 

9|=9T42  tional  parts,  |  and  j,  to  a  common  de- 

2|=:2-97y  nominator  12.     Since  we  cannot  take 

^  from  T42,  we  add  1— i|  to  T4-j,  which 

6T5  J.TIS.        makes  I£,  and  y9^  from  1^  leaves  T"^. 

We  now  add  1  to  the  2  in  the  subtrahend,   and  say,  3  from 

9  leaves  6.     We  thus  obtain  6^,  the  difference  required. 

Hence,  to  subtract  mixed  numbers,  we  may  reduce  the 
fractional  parts  to  a  common  denominator,  and  then  subtract 
the  fractional  and  integral  parts  separately. 

19.  From  24|  take  174.  Ans,  7f 

20.  From  147|  take  49}.  Ans.  98T53. 

21.  From  75^  take  40|.  Ans.  3411. 

22.  From  63T%  take  22|.  Ans.  40f. 

23.  Bought  flour  at  6|  dollars  a  barrel,  and  sold  it  at  7| 
dollars  a  barrel ;  what  was  the  gain  per  barrel  ? 

Ans.  T9Q  of  a  dollar. 

24.  From  a  cask  of  wine  containing  38|  gallons,  15|  gal- 
lons were  drawn  ;  how  many  gallons  remained  ? 

Am.  22  £|  gallons 


88  FRACTIONS. 

MULTIPLICATION". 
CASE  I. 

97*  To  multiply  a  fraction  by  an  integer. 

1.  If  1  pound  of  sugar  cost  $  of  a  dollar,  how  much  will 
3  pounds  cost  ? 

ANALYSIS.    If  1  pound  cost  i  of  a  dollar,  3  pounds,  which 
are  3  times  1  pound,  will  cost  3  times  ^  or  |  of  a  dollar.     There- 

fore, 3  pounds  of  sugar,  at  ^  of  a  dollar  a  pound,  will  cost  j*  of 
a  dollar. 

2.  If  1  horse  eat  |  of  a  ton  of  hay  in  1  month,  how  much 
will  4  horses  eat  in  the  same  time  ? 

3.  At  |  of  a  dollar  a  bushel,  what  will  be  the  cost  of  2 
bushels  of  pears  ?  of  3  bushels  ?  of  5  bushels  ? 

4.  How  many  are  3  times  f  ?  5  times  |  ?  4  times  J  ? 
6  times  §  ?  9  times  y\j  ?  8  times  f  ? 

5.  If  one  yard  of  cloth  cost  |  of  a  dollar,  how  much  will 
3  yards  cost? 

FIRST  OPERATION.  ANALYSIS.     In  the  first  operation  we 

|X3=-g5-=2^.  multiply  the  fraction  by  3,  by  multi- 

SECOND  OPERAT!ON.  P1?^  its  numerator    b7  3»  obtaining 

_  5  _  01  ~5"==^  ^  •     ^n  *n*s  case  ^ne  'Da^ue  of  the 


fractional  unit  remains  the  same,  but 
we  multiply  the  number  taken,  8  times.  In  the  second  opera- 
tion, we  multiply  the  fraction  by  3,  by  dividing  its  denominator 
by  3,  obtaining  |  =  2J.  In  this  case,  the  value  of  the  fractional 
unit  is  multiplied,  8  times,  but  the  number  taken,  is  the  same. 
Hence, 

Multiplying  a  fraction  consists  in  multiplying  it*  nu- 
merator, or  dividing  its  denominator. 

NOTK.    Always  divide  the  denominator  when  ft  is  exactly  divisible  by  iue  multi- 
plier. 


MULTIPLICATION.  89 

EXAMPLES   FOR   PRACTICE. 

6.  Multiply  «  by  5.  Ans.  4f . 

7.  Multiply  J  by  4.  Ans.  3  J. 

8.  Multiply  T*  by  6.  ^TW.  5f . 

9.  Multiply  4f  by  9.  Ans.    4. 

10.  Multiply  }?  by  3.  ^ns.  1J. 

11.  Multiply  |f  by  14.  Ans.  10. 

12.  Multiply  4|  by  5. 

OPERATION.  ANALYSIS.     In  multiplying  a 

4j  mixed  number,  we  first  multiply 

5  .          the  fractional  part,  then  the  inte- 

—         Or,  ger,  and  then  add  the  two  pro- 

If     4|  =  Y  ducts.     Thus,  5  X  i  =  -I  =  If  ; 

20       ^X5=¥=21f      and  5x4  =  20,  which  added  to 
ITT  If,  gives  21|,  the  required  re- 

sult.     Or,  we  may  reduce  the 
mixed  number  to  an  improper  fraction,  and  then  multiply  it. 

13.  Multiply  6|  by  8.  Ans.  54. 

14.  Multiply  9|  by  7.  Ans.  68f . 
.15.  If  a  man  earn  1|  in  1  day,  how  much  will  he  earn  in 

10  days  ?  Ans.  18  f  dollars. 

16.  What  will  14  yards  of  cloth  cost,  at  f  of  a  dollar  a 
yard  1  Ans.  10  dollars. 

17.  At  3£  dollars  a  cord,  what  will  be  the  cost  of  20 
cords  of  wood  ?  Ans.  65  dollars. 

18.  If  one  man  can  mow  Ij9^  acres  of  grass  in  a  day,  how 
many  acres  can  5  men  mow?  Ans.  9£  acres. 

19.  What  will  9  dozen  eggs  cost,  at  14  £  cents  a  dozen  ? 

Ans.  130  J  cents. 

20.  At  64  dollars  a  barrel,  what  will  30  barrels  of  flour 

««    ' 

cost?  Am.     204  dollars. 


90  FRACTIONS. 

CASE  II. 

98.  To  multiply  an  integer  by  a  fraction. 

1.  At  9  dollars  a  barrel,  what  will  |  of  a  barrel  of  flour 
cost? 

ANALYSIS.  Since  1  barrel  of  flour  cost  9  dollars,  f  of  a  barrel 
will  cost  2  times  £  of  9  dollars.  £  of  9  dollars  is  3  dollars,  and 
|  of  9  dollars  is  2  times  8  dollars,  or  6  dollars.  Therefore  £  of 
a  barrel  will  cost  6  dollars. 

2.  If  a  yard  of  cloth  be  worth  8  dollars,  what  is  |  of  a 
yard  worth  1 

3.  If  an  acre  of  land  produce  25  bushels  of  wheat,  how 
much  will  \  of  an  acre  produce  ?   f  of  an  acre  1   |  of  an 
acre  ? 

4.  If  a  man  earn  20  dollars  in  a  month,  how  much  can  he 
earn  in  £  of  a  month  ?  in  f  1  in  -^  ?  in  |  ? 

5.  If  a  ton  of  hay  cost  12  dollars,  how  much  will  ^  of  a 
ton  cost  ?  |  of  a  ton  ?  f  of  a  ton  ?  |  of  a  ton  ? 

6.  At  60  dollars  an  acre,  what  will  |  of  an  acre  of  land 
cost? 

FIRST  OPERATION.  ANALYSIS.     4  fifths  of  an  acre 

5)60   Price  of  1  acre-  will  cost  4  times  as  much  as  1  fifth 

T2    cost  of  |  of  an  acre.         of  an  acre,  or  4  times  ^  of  60  dol- 

4  lars.     \  of  60  dollars  is  12  dollars, 

and  4  is  4  times  12,  or  48  dollars, 

4o    cost  of  *  of  an  acre.  6 

the  oost  of  |  of  an  acre.  In  the 

BECOND   OPERATION.  gecond  operation>  we  multiply  the 

60    price  of  1  acre.  price  of   j   acre  by  ^  afid  obtain 

240  dollars,  the  cost  of  4  acres ; 

5)240    cost  of  4  acres. 

L —  but  as  I  of  1  acre  is  the  same  as 

48    cost  of  4  of  an  acre,        t      „     ' 

\  of  4  acres,  we  divide  240  dol- 
lars, the  cost  of  4  acres,  by  5,  and  obtain  48  dollars,  the  cost  of 
of  of  acre.  Hence, 


MULTIPLICATION.  9J 

RULE.  Multiplying  an  integer  ly  a  fraction,  consists  in 
multiplying  by  the  numerator,  and  dividing  the  product  by 
the  denominator. 

7.  Multiply  45  by  |.  Ans.  33f . 

8.  Multiply  68  by  £.  Ans.  54f . 

9.  Multiply  105  by  T76.  Ans.    49. 

10.  Multiply  480  by  f .  Ans.  300. 

11.  At  16  dollars  a  ton,  what  will  be  the  cost  of  j  of  a 
ton  of  hay  ?  Ans.  12  dollars. 

12.  If  a  village  lot  is  worth  340  dollars,  what  is  f  of  it 
worth  ?  Ans.  255  dollars. 

13.  If  a  hogshead  of  sugar  is  worth  75  dollars,  what  is 
l£  of  it  worth  ?  Ans.  68|  dollars. 

14.  If  an  acre  of  land  produce  114  bushels  of  oats,  how 
many  bushels  will  T9g  of  an  acre  produce  ? 

Ans.  64|  bushels. 

15.  If  a  man  travel  47  miles  in  a  day,  how  far  does  he 
travel  in  f  of  a  day  ?  Ans.  29|  miles. 

CASE   III. 

99.  To  multiply  a  fraction  by  a  fraction. 

1.  If  a  bushel  of  apples  is  worth  |  of  a  dollar,  what  is  \ 
of  a  bushel  worth  ? 

ANALYSIS.  Since  1  bushel  is  worth  \  of  a  dollar,  \  of  a  bush- 
el is  worth  \  times  \  of  a  dollar ;  £  equals  f ,  and  a  \  of  f  is  £. 
Therefore  \  of  a  bushel  is  worth  \  of  a  dollar. 

2.  If  a  yard  of  cloth  cost  A  a  dollar,  how  much  will  \  of 
a  yard  cost  ? 

3.  When  oats  are  worth  J  of  a  dollar  a  bushel,  what  is  | 
of  a  bushel  worth. 

4.  If  a  man  own  4  of  a  vessel,  and  he  sells  \  of  his  share 
what  part  of  the  vessel  does  he  sell  ? 


92  FRACTIONS. 

5.  At  |  of  a  dollar  a  bushel,  what  will  £  of  a  bushel  of 
corn  cost  ? 

OPERATION.  ANALYSIS.    Since  1  bushel  cost 

|Xf=14T=2  Ans.  |  of  a  dollar,  |  of  a  bushel  will 
cost  |  times  |  of  a  dollar.  By  multiplying  the  numerators  2 
and  3  together,  we  obtain  the  numerator  6  of  the  product ;  and 
by  multiplying  the  denominators  8  and  4  together,  we  obtain 
the  denominator  12  of  the  product,  and  thus  we  have  -^=^  for 
the  required  product.  Hence  we  have  the  following 

RULE.  Multiply  together  the  numerators  for  a  new  nu- 
merator',  and  the  denominators  for  a  new  denominator,  and 
reduce  the  result  to  its  lowest  terms. 

EXAMPLES    FOR  PRACTICE. 

6.  Multiply  4  by  f .  Ans.  TV 

7.  Multiply  |  by  f .  Ans.  295. 

8.  Multiply  |  by  f .  Ans.  f  £. 
-     9.  Multiply  |  by  f .  Ans.  2\. 

10.  Multiply  -?2  by  f .  Ans.  T62. 

11.  What  is  the  product  of  f ,  |  and  f  ?  Ans.  ^. 

12.  What  is  the  product  of  f ,  |  and  f  ?  Ans.  ?%. 

13.  What  is  the  product  of  J,  f  and  ^  ?  Ans.  -|. 

14.  What  is  the  product  of  £f  and  £§  ?  ^Ins.  j. 

15.  What  is  the  product  of  f ,  1|,  5  and  j  ? 

OPERATION.  When  integers  or  m?'#- 

|XlAX^X|=  ec^  numbers  occur  among 

|X  jXf  Xf^V^S  ^4ns.  the  given  factors,  they 

may  be  reduced  to  improper  fractions  before  multiplying ; 

and  an  integer  may  be  reduced  to  the  form  of  a  fraction  by 

writing  1  for  its  denominator ;  thus  5=f . 

16.  What  is  the  product  of  f ,  f  and  2f  ?         Ans.  Jf . 

17.  What  is  the  product  of  3,  T9a  and  |  ?        Ans.  2f 


MULTIPLICATION.  93 


18.  What  is  the  product  of  |,  T5T  and  f  f  ? 

19.  Find  the  value  of  f  of  f  multiplied  by  f  o 

OPERATION. 


NOTES.  1.  Fractions  with  the  word  of  between  them  are  sometimes  called  com,' 
pound  fractions.  The  word  of  is  simply  an  equivalent  for  the  sign  of  multiplica- 
tion, and  signifies  that  the  numbers  between  which  it  is  placed  are  to  be  multiplied 
together. 

2.  When  the  same  factors  occur  in  both  numerator  and  denominator  of  fractions 
to  be  multiplied  together,  they  may  be  omitted  and  the  remaining  factors  only 
used;  thus,  5  and  3  being  found  in  both  the  numerators  and  denominators  of  the 
above  example  may  be  omitted  in  multiplying. 

20.  Multiply  |  of  f  by  |  of  £.  Ans.  Ji- 

21.  Multiply  |  of  3  by  |  of  2^.  Ans.  5|. 

22.  What  is  the  product  of  T%,  ^  of  f  and  \±  1 

Ans.  ^. 
^  23.  What  is  the  product  of  f  of  T7T  by  54  1      Am.  3. 

24.  What  is  the  value  of  f  times  £  of  f  of  10  ? 

Ans.  | 

25.  What  is  the  value  of  T52  of  f  times  \  of  3  f  ? 

Ans.  f. 

26.  At  |  of  a  dollar  a  bushel,  what  will  |  of  a  bushel  of 
corn  cost  1  .  Ans.  ±  of  a  dollar. 

27.  When  peaches  are  worth  T9^  of  a  dollar  a  .bushel, 
what,  is  |  of  a  bushel  worth?  Ans.  ^  dollar. 

28.  Jane  having  |  of  a  yard  of  silk  gave  |  of  it  to  her 
sister  ;  what  part  of  a  yard  did  she  give  her  sister  ? 

Ans.  |  of  a  yard. 

29.  When  pears  are  worth  J  of  a  dollar  a  basket,  what  is 
^  of  |  of  a  basket  worth  ?  Ans.  |  of  a  dollar. 

30.  A  man  owning  ^  of  a  ship,  sold  |  of  his  share; 
what  part  of  the  whole  ship  did  he  sell  ?  'Ans.  -||. 

31.  A  grocer  having  ^f  of  a  hogshead  of  molasses  sold 
£$  of  it  ;  what  part  of  a  hogshead  remained  1 

32.  At  §  of  a  dollar  a  yard,  what  will  be  the  cost  of  -i  of 
8  yards  of  cloth  1  Ans.  U  dollars. 


94  FRACTIONS. 

DIVISION. 
CASE  I. 

IOO.  To  divide  a  fraction  by  an  integer. 

1.  If  3  pounds  of  raisins  cost  5  of  a  dollar,  how  much 
will  1  pound  cost  ? 

ANALYSIS.  If  3  pounds  cost  £  of  a  dollar,  1  pound  which  is 
£  of  3  pounds,  will  cost  £  of  £,  or  £  of  a  dollar.  Therefore,  1 
pound  will  cost  £  of  a  dollar. 

2.  If  4  pounds  of  coffee  cost  ^  of  a  dollar,  how  much  will 
1  pound  cost  ? 

3.  If  5  marbles  cost  |  of  a  dollar,  how  much  will  1  mar- 
ble cost  ? 

4.  If  J  of  a  barrel  of  flour  be  equally  divided  among  6 
persons,  what  part  of  a  barrel  will  each  have  ? 

5.  If  4  of  a  box  of  tea  be  equally  distributed  among  8 
persons,  what  part  of  a  box  will  each  have  ? 

6.  Paid  f  of  a  dollar  for  4  pounds  of  butter ;  what  was 
the  cost  per  pound  ? 

FIRST  OPERATION.  ANALYSIS.     In  the  first  operation 

!-j-4=§  Ans.         we  divide  the  fraction  by  4,  by  divid- 
ing its  numerator  by  4,  obtaining  |. 

SECOND  OPERATION.       In  this  case  the  value  of  the  frartional 

|-j-4=38g=§  Ans.     unit  is  unchanged,  but  we  diminish 

the  number  taken^  4  times.     Ii  the 

second  operation  we  divide  the  fraction  by  4,  by  multiplying 
the  denominator  by  4,  obtaining  ^8ff==|.  In  this  case  the  val- 
ue of  the  fractional  unit  is  diminished  4  times,  but  the  number 
taken  is  the  same.  Hence, 

Dividing  a  fraction  consists  in  dividing  its  numerator,  or 
multiplying  its  denominator. 

NOTB.  We  divide  the  numerator  vrhen  it  is  exactly  divisible  by  the  divisor^  oth- 
erwise we  multiply  the  denominator 


DIVISION.  95 

EXAMPLES   JFOB  PRACTICE. 

7.  Divide  Jj  by  3.  Ans.  §. 

8.  Divide  |  by  4.  Ans.  J. 

9.  Divide  j  j  by  5.  ^4/is.  T25. 

10.  Divide  i|  by  5.  Ans.  T3e. 

11.  Divide  |  by  9.  ^dws.  6»a. 

12.  Divide  |§  by  21.  Ans.  fo 

13.  Divide  |  of  f  by  12.  ^s.  T'ff . 

14.  Divide  |  of  f  by  6.  Ans.  -fa. 

15.  Divide  4|  by  7. 

OPERATION. 

4^= 2^  NOTE.    We  reduce  the  mixed  num- 

2gi-i-7=f  -<4.?is.      ber  to  an  improper  fraction  and  then 

divide  as  before. 

16.  Divide  3|  by  4.  AM.  JJ. 

17.  Divide  6j  by  9.  Ans.  %%. 

18.  Divide  4  of  2^  by  3.  Ans.  j. 

19.  Divide  8^  by  12.  ^Ins.  f  j. 

20.  Divide  13 j  by  10.  Ans.  If. 

21.  Divide  |  of  8  by  20.-  Ans.  J. 

22.  If  6  persons  agree  to  share  equally  |  of  a  bushel  of 
grapes,  what  part  of  a  bushel  will  each  have  ?       Ans.  |. 

23.  If  5  yards  of  sheeting  cost  T9<j  of  a  dollar,  what  will 
JL  yard  cost  ?  Ans.  -f^  of  a  dollar. 

24.  If  8  bushels  of  apples  cost  5|  dollars,  what  will  1 
Dushel  cost  ?  Ans.  |  of  a  dollar. 

25.  If  J  of  10  pounds  of  butter  cost  l\ Collars,  what 
will  1  pound  cost  ?  Ans.  |  of  a  dollar. 

26.  A  man  distributed  J$  of  a  dollar  equally  among  6 
beggars ;  what  part  of  a  dollar  did  he  give  to  each  ? 

27.  If  f  of  9  cords  of  wood  cost  12|  dollars,  what  will 
1  oord  cost] 


96  FRACTIONS. 

CASE  II. 

•  1O1.  To  divide  an  integer  by  a  fraction. 

1.  At  j   of  a  dollar  a  yard,  how  many  yards  of  ribbon 
can  be  bought  for  2  dollars? 

ANALYSIS.  As  many  yards  as  |  of  a  dollar,  the  price  of  1 
yard,  is  contained  times  in  2  dollars.  Since  in  1  dollar  there 
are  3  thirds  of  a  dollar,  in  two  dollars,  there  are  2  times  3  thirds, 
or  G  thirds ;  and  1  third  is  contained  in  6  thirds,  6  times. 
Therefore  6  yards  of  ribbon  can  be  bought  for  2  dollars. 

2.  When  potatoes  are  |  of  a  dollar  a  bushel,  how  n.any 
bushels  can  be  bought  for  2  dollars  ?  for  4  dollars  ?  for  6 
dollars  ? 

3.  If  a  man  spend  |  of  a  dollar  a  day  for  cigars,  how  long 
will  it  take  him  to  spend  3  dollars  ?  5  dollars  ?  6  dollars  9 

4.  At  |  of  a  dollar  a  bushel,  how  many  bushels  of  corn 
can  be  bought  for  16  dollars  1 

JIRST  OPERATION.  ANALYSIS.     As  many  bushels  as  J  of 

a  dollar,  the  price  of  1  bushel,  is  con- 
tained tira»s  in  16  dollars.     But  we  can- 
not  divide  integers  by  fifths,  because 
they  are  not  of  the  same  denomination. 
20  bushels.       Reducing  16  dollars  to  fifths  by  multi 
SECOND  ^OPERATION,     plying  by  5,  we  have  80  fifths,  and  4 
4)14  fifths  is  contained  in  SO  fifths,  20  times, 

>        r  the  required  number  of  bushels.      In 

the  second  operation,  we  divide  the  in- 

20  bushels.  teger  by  the  numerator  of  the  fraction, 
and  multiply  the  quotient  by  the  denominator,  which  produces 
the  same  result  as  in  the  first  operation.  Hence 

Dividing  "by  a  fraction  consists  in  multiplying  by  the 
denominator,  and  dividing  the  product  by  the  numerator 
of  the  divisor. 


DIVISION. 


97 


EXAMPLES   FOR   PRACTICE. 


6.  Divide    18  by  f . 

6.  Divide    14  by  f . 

7.  Divide    11  by  f . 

8.  Divide    75  by  •£$. 

9.  Divide  120  by  T6T. 

10.  Divide    96  by  {?. 

11.  Divide  226  by  &. 

12.  Divide    28    by  4|. 

OPERATION. 

28X3=84 
84-4-14=6  Ans. 


Ans.  27. 

Am.  49. 

AMS.  19f. 

Ans.  83|. 

Ans.  220. 

Ans.    186. 

Ans,   627J. 


13.  Divide    16  by 

14.  Divide    42  by 

15.  Divide  112  by 

16.  Divide  180  by 

17.  Divide  425  by 

18.  Divide  318  by 


ANALYSIS.  We  reduce  the  mixed 
number  to  an  improper  fraction,  and 
then  divide  the  integer  in  the  same 
manner  as  by  a  proper  fraction. 

21.  Ans.       7£. 

3A.  Ans.       12. 

6|.  Ans.       17^. 

7|.  Ans.      25  -fy. 

f  Ans.    595. 

2V  Ans.  1219. 

19.  When  potatoes  are  ^  of  a  dollar  a  bushel,  how  many 
bushels  can  be  bought  for  10  dollars  ?        Ans.  12 «  bush. 

20.  Divide  9  bushels  of  corn  among  some  persons,  giving 
them  T3g  of  a  bushel  each ;  how  many  persons  will  receive 
a  share?  Ans. 48. 

21.  At   2 1  dollars  a  cord,  how  many  cords  of  wood  can 
be  bought  for  27  dollars  ?  Ans.  9T9?  cords. 

22.  If  a   horse  eat   |   of  a  bushel  of  oats  in  a  day,  in 
how  many  days  will  he  eat  20  bushels  ?       Ans.  36  days. 

23.  If  a  man  walk  2T9<j  miles  an  hour,  how  many  hours 
will  he  require  to  walk  48  miles  ?  Ans.  16^|  hours. 

24.  At  Jg  of  a  dollar  a  pound,  how  many  pounds  of  rice 
can  be  bought  for  3  dollars  7  Ans.  48  pounds. 


98  FRACTIONS. 

CASE  III. 

1O2.  To  divide  a  fraction  by  a  fraction. 

1.  At  |  of  a  dollar  a  pound,  how  many  pounds  of  tea  can 
be  bought  for  J  of  a  dollar  ? 

ANALYSIS.  As  many  pounds  as  |  of  a  dollar,  the  price  of  1 
pound,  is  contained  times  in  |  of  a  dollar ;  2  fifths  are  contain- 
ed in  4  fifths,  2  times.  Therefore  2  pounds  can  be  bought  for 
J  of  a  dollar. 

Hence  we  see,  that  when  fractions  have  a  common  denomina- 
tor, division  may  be  performed  by  dividing  the  numerator  of 
the  dividend  by  the  numerator  of  the  divisor. 

2.  How  many  pine-apples  at  T30  of  a  dollar  each,  can  be 
bought  for  T6jj  of  a  dollar?  for  y^?  for  }|? 

3.  If  a  horse  eat  ^  of  a  bushel  of  oats  in  1  day,  in  how 
many  days  will  he  eat  |  of  a  bushel  If?   y>  ?   \*  ? 

4.  At  ^  of  a  dollar  a  bushel,  how  many  bushels  of  ap- 
ples can  be  bought  for  f  of  a  dollar  ?  for  ^  ?  for  f  ? 

5.  At  |  of  a  dollar  a  pound,  how  many  pounds  of  tea  can 
be  bought  for  |  of  a  dollar  ? 

FIRST  OPERATION.  ANALYSIS.  As  many  pounds 

§=3%;  |=2U-  as  I  of  a  dollar,  the  price  of 

io'^slj— IB  Ans.  1  pound,  is  contained  times 

SECOND  OPERATION  in  |  of  a  dollar.     f  equal 

£-*-f=:|Xf  =  g5=1B  Ans-  28<J»  i  e(lual  if'  and  8  twenti- 
eths are  contained  in  15  twen- 
tieths 1|  times.  Or,  as  in  the  second  operation,  we  have  multi- 
plied the  dividend  |,  by  the  denominator  5,  of  the  divisor,  and  di- 
vided the  result  by  the  numerator  2,  of  the  divisor.  Iience,by 
inverting  the  terms  of  the  divisor  the  two  fractions  will  stand  in 
§uch  relation  to  each  other,  that  we  can  multiply  together  the 
two  upper  numbers  for  the  numerator  of  the  quotient,  and  the 
two  lower  numbers  for  the  denominator,  as  shown  in  the  second 
operation.  Hence  * 


DIVISION.  99 

* 

KULE.  I.  Reduce  integers  and  mixed  numbers  to  im- 
proper fractions. 

II.  Invert  the  terms  of  the  divisor,  and  proceed  as  in  mul- 
tiplication. 

EXAMPLES   FOR   PRACTICE. 

6.  Divide  T%  by  T%.  Ans.  3. 

7.  Divide  ^  by  \.  Ans.  2. 

8.  Divide  f  by  f .  Ans.  jf . 

9.  Divide  J  by  f .  Ans.  2T3g. 

10.  How  many  times  is  T7^  contained  in  -}  J  ?  Ans.  2|. 

11.  How  many  times  is  f  contained  in  |  ?       Ans.  -fa. 

12.  How  many  times  is  A  contained  in  j|  ?     ^4ns.  1|. 

13.  Divide  |  oi'|  by  f.  Ans.  f 

14.  Divide  f  of  f  by  &.  Ans.  If 

15.  Divide  ji  by  4  of  f.  .          Ans.  7TV 

16.  Divide  |  of  £  by  f  of  J.  ^ns.  1T^. 

17.  At  ^  of  a  dollar  a  pound,  how  many  pounds  ot  su- 
gar can  be  bought  for  |  of  a  dollar  ?        Ans.  5|  pounds. 

18.  At  T7^  of  a  dollar  a  pint,  how  much  wine  can  be 
bought  for  \  of  a  dollar  ?  Ans.  f  of  a  ^pint. 

19.  At  -|  of  I  of  a  dollar  a  yard,  how  many  yards  of  rib- 
bon can  be  bought  for  T7^  of  a  dollar  ?         Ans.  2|  yards. 

20.  At  ^  of  a  dollar  a  yard,  how  many  yards  of  silk  can 
be  bought  for  |  of  a  dollar  ?  Ans.  2|  yards. 

21.  A  man  owning  f  of  a  copper  mine,  divided  his  share 
equally  among  his  sons,  giving  them  T6ff  each ;  how  many 
sons  had  he  ?  Ans.  2. 

22.  If  ^  of  a  bushel  of  pears  cost  |  of  a  dollar,  how  much 
will  1  bushel  cost  1  Ans.  ^  of  a  dollar. 

23.  How  much  corn  at  $  of  a  dollar  a  bushel,  can  be 
bought  for  |  of  a  dollar.  Ans.  {  of  a  bushel. 


100  FRACTIONS. 

^ 

PROMISCUOUS   EXAMPLES. 

1.  In  25T9ff  pounds  how  many  IGtlis  of  a  pound] 

2.  Reduce  ^V  to  a  mixed  number.  Ans.  11||. 

3  o  o  u 

3.  Reduce  ^{jj  to  its  lowest  terms.  Ans.     j. 

4.  In  7  855  9  of  a  day  how  many  days  1 

5.  Change  42  pounds  to  sevenths  of  a  pound. 

6.  Reduce  21  i  to  an  improper  fraction,       ylns.    J|9. 

7.  Reduce  126|  to  thirds.  Ans.  -£f-&. 

8.  Reduce  1 1§  to  its  lowest  terms.  Ans.  |. 

9.  Reduce  4  and  |  to  a  common  denominator. 

10.  Reduce  36  to  a  fraction  whose  denominator  is  12. 

11.  What  is  the  sum  of  |,  |  and  {  1  Ans.  If. 

12.  Add  together  T9^,  ^  and  34.  Ans.  4|. 

13.  What  is  the  difference  between  f  and  |  ? 

14.  Reduce  T9ff,  f  and  |  to  a  common  denominator. 

15.  Sold  9f  cords  of  wood  to  one  inan;  and  12T9g  to  an- 
other ;  how  much  did  I  sell  to  both  ] 

16.  Paid  87T9o  dollars  for  a  horse,  and  62^  dollars  for  a 
wagon ;  how  much  more  was  paid  for  the  horse  than  the 
wagon  ?  Ans.  25|  dollars. 

17.  A  farmer  having  234{|  acres  of  land,  sells  at  one 
time  42|   acres,  at  another  time  61|,  and  at  another  70^- 
asres ;  how  many  acres  has  he  left  ?       Ans.  60T6g  acres. 

18.  A  speculator  bought  120  bushels  of  wheat,  for  136| 
dollars,  and  sold  it  for  197 1  dollars;  how  much  did  he  gain? 

19.  Bought  12  pounds  of  coffee  at  ^  of  a  dollar  a  pound, 
and  9  pounds  of  tea  at  J  of  a  dollar  a  pound;  what  was  the 
cost  of  the  whole  ?  Ans.  Sy7^  dollars. 

20.  Bought  10  bushels  of  wheat,  at  1^  dollars  a  bushel, 
and  14  bushels  of  corn,  at  |  of  a  dollar  a  bushel ;  which 
cost  the  more,  and  how  much  ? 

Ans.  the  corn,  3 \  dollars. 


PKOMISCUOUS   EXAMPLES.  101 

21.  Paid  12  dollars  for  some  cloth,  at  the  rate  of  J  of  a 
lollar  a  yard ;  how  many  yards  was  purchased  ? 

22.  If  8  oranges  cost  f  of  14  dollars,  what  will  1  orange 
3ost  ?  Ans.  yL  of  a  dollar. 

23.  A  man  bought  J  of  a  farm  and  sold  f  of  his  share ; 
what  part  of  the  whole  farm  did  he  sell  ?  what  part  had  he 
eft1?  Am.  Sold  44. 

24.  If  a  barrel  of  sugar  is  worth  22  dollars,  what  is  T7^  of 
it  worth  ?  Ans.  15  f  dollars. 

25.  How  many  hours  will  it  take  a  man  to  travel  136 
miles,  if  he  travel  3|  miles  an  hour  ?     Ans.  4129g  hours. 

26.  How  many  barrels  of  apples  can  be  bought  for  18 
dollars,  at  !T3g  dollars  a  barrel  1  Ans.  15  T3^  barrels. 

27.  If  the  smaller  of  two  fractions  be  T4T,  and  the  differ- 
ence f ,  what  is  the  greater  ]  Ans.  ||. 

28.  If  the  sum  of  two  fractions  is  1|,  and  one  of  them  is 
5»3,  what  is  the  other  1  Ans.  f  J. 

29.  If  the  dividend  be  f  f,  and  the  quotient  f ,  what  is 
the  divisor?  Ans.  1^. 

30.  If  the  divisor  be  T9g,  and  the  quotient  3  J,  what  is  the 
dividend  ?  Ans.  2T45. 

31.  How  many  bushels  of  oats  worth  |  of  a  dollar  a  bush- 
el, will  pay  for  f  of  a  barrel  of  flour  worth  9  dollars  a  bar- 
rel 1  Ans.  15  bushels. 

32.  At  |  of  a  dollar  a  rod,  what  will  it  cost  to  dig  J  of  | 
of  5^  rods  of  ditch  ?  Ans.  T9T93  dollars. 

33.  If  a  man  has  24  J  bushels  of  clover  seed,  and  he  sells 
|  of  it,  how  much  has  he  left  ?  Ans.  Q^  bushels. 

34.  A  man  had  6  lots  of  land,   each   containing  37| 
acres  ;  how  many  acres  did  they  all  contain  ? 

35.  If  |  of  a  ton  of  hay  can  be  bought  for  15  dollarsu 
what  part  of  a  ton  can  be  bought  for  1  dollar? 


102  DECIMALS. 


DECIMAL  FRACTIONS. 

NOTATION   AND   NUMERATION. 

1O3.  Decimal  Fractions  are  fractions  which  have  for 
their  denominator  10,  100,  1000,  or  1  with  any  number  of 
ciphers  annexed. 

Decimal  fractions  are  commonly  called  decimals. 

Since  TV=TV$y,  Tiu=T<5oo>  &c->  tlie  denominators  of 
decimal  fractions  increase  and  decrease  in  a  tenfold  ratio, 
the  same  as  simple  numbers. 

1  04:.  In  the  formation  of  Decimals  a  unit  is  divided  in- 
to 10  equal  parts,  called  tenths  ;  each  of  these  tenths  is  di- 
vided into  10  other  equal  parts  called  hundredths  ;  each  of 
these  hundredths  into  10  other  equal  parts,  called  thou- 
sandths j  and  soon.  Since  the  denominators  of  decimal 
fractions  increase  and  decrease  by  the  scale  of  10,  th-j  same 
as  simple  numbers,  in  writing  decimals  the  denominators 
may  be  omitted. 

1O5.  The  Decimal  sign  (.)  is  always  placed  before  deci- 
mal figures  to  distinguish  them  from  integers.     It  is  com- 
monly called  the  decimal  point.     Thus, 
T6£     is  expressed  .6 


"        -279 

.5       is  5  tenths,  which  =  y1^  of  5  units  ; 

.05     is  5  hundredths,  "       =tff  °^  «*  tenths; 

.005  is  5  thousandths,  "      =T\y  of  5  hundredths. 

And  universally,  the  value  of  a  figure  in  any  decimal 
place  is  T\j  the  value  of  the  same  figure  in  the  next  left 
bond  place. 


NOTATION  AND  NUMERATION.  103 

1OO.  The  relation  of  decimals  and  integers  to  each  oth- 
er is  clearly  shown  by  the  following 

DECIMAL   NUMERATION   TABLE. 


5732754.573256 
By  examining  this  table  we  see  that 

Tenths  are  expressed  by  one  figure. 

Hundredths          "          "          "   two  figures. 
Thousandths         "          "          "   three    " 
1O7.  Since  the  denominator  of  tenths  is  10,  of  hun- 
dredths  100,  of  thousands  1000,  and  so  on,  a  decimal  may 
be  expressed  by  writing  the  numerator  only  ;  but  in  this 
case  the  numerator  or  decimal  must  always  contain  as  many 
decimal  places  as  are  equal  to  the  number  of  ciphers  in  the 
denominator  ;  and  the  denominator  of  a  decimal  will  al 
ways  be  the  unit,  1,  with  as  many  ciphers  annexed  as  are 
equal  to  the  number  of  figures  in  the  decimal  or  numerator, 
The  decimal  point  must  never  be  omitted. 


EXAMPLES  FOR  PRACTICE. 

1.  Express  in  figures  seven-tenths.  Ans.  .7. 

2.  Write  twenty-five  hundredths.  Ans.  .25. 
8.  Write  nine  hundredths.                               Ans.  .09. 

4.  Write  one  hundred  twenty-five  thousandths. 

5.  Write  eighteen  thousandths. 


104  DECIMALS. 

6.  Write  fifty-eight  hundredths. 

7.  Write  two  hundred  thirty-six  thousandths. 

8.  Write  one  thousand  three  hundred  twenty  ten-thou- 
sandths. Am.  .1320. 

9.  Write  seven  hundred  thirty-two  ten-thousandths. 

Read  the  following  decimals  : 

.06  .143  .000  .479 

.84  .037  .3240  .00341 

.80  .472  .1026  .102367 

1O8.  A  mixed  number  is  a  number  consisting  of  inte- 
gers and  decimals ;  thus,  71.406  consists  of  the  integral 
part,  71,  and  the  decimal  part,  .406 ;  it  is  read  the  same  as 
71T4<j°<&,  71  an(i  406  thousandths. 

EXAMPLES    FOE  PRACTICE. 

1.  Write  twenty-four,  and  four  tenths.         Ans.  24.4. 

2.  Write  thirty-two,  and  five  hundredths, 

3.  Write  seventy-six,  and  forty-six  thousandths. 

4.  Write  one  hundred  twelve,  and  one  hundred  ninety 
thousandths.  Ans.  112.190. 

5.  Write  sixty-three,  and  forty-four  ten-thousandths. 

6.  Write  seventy-five,  and  one  hundred  forty  ten-thou- 
sandths. 

7.  Write  five,  and  5  hundred  thousanths. 

8.  Write  sixteen,  and  21  ten-thousan^tis. 

9.  Write  eight,  and  234  hundred  thousai 

10.  Write  forty,  and  75  hundred  thousandths.^ 

Ans.  40.00075. 

11.  Read  the  following  numbers  : 

42.08  50.002  640.00010 

81.110  161.0301  7.4230 

120.0342  14.42000  3.01206 


NOTATION  AND  NUMERATIO 

1OO.  From  the  foregoing  explanations  and  illustrations 
we  derive  the  following  important 

PRINCIPLES  OF  DECIMAL  NOTATION  AND  NUMERATION. 

1.  The  value  of  any  decimal  figure  depends  upon  its 
place  from  the  decimal  point  j  thus  .3  is  ten  times  .03. 

2.  Prefixing  a  cipher  to  a  decimal  decreases  its  value  the 
same  as  dividing  it  by  ten ;  thus,  .03  is  ^  the  value  of  .3. 

3.  Annexing  a  cipher  to  a  decimal  does  not  altar  its  val- 
ue, since  it  does  not  change  the  place  of  the  significant  fig- 
ures of  the  decimal ;  thus,  T60,  or,  .6,  is  the  same  as  y6^,  or 
.60. 

4.  Decimals  increase  from  right  to  left,  and  decrease  from 
left  to  right,  in  a  tenfold  ratio ;  and  therefore  they  may  be 
added,  subtracted,  multiplied,  and  divided  the  same  as  whole 
numbers. 

5.  The  denominator  of  a  decimal,  though  never  express- 
ed, is  always  the  unit   1,  with  as  many  ciphers  annexed  as 
there  are  figures  in  the  decimal. 

6.  To  read  decimals  requires  two  numerations;  first,  from 
units,  to  find  the  name  of  the  denominator,  and  second,  to- 
wards units,  to  find  the  value  of  the  numerator. 

1 1 0.  Having  analyzed  all  the  principles  upon  which 
the  writing  and  reading  of  decimals  depend,  we  will  now 
present  these  principles  in  the  form  of  rules. 

RULE   EOR   DECIMAL   NOTATION. 

I.  Write  the  decimal  the  same  as  a  whole  number,  placing 
ciplvers  where  necessary  to  give  each  significant  figure  its  true 
local  value. 

II.  Place  the  decimal  point  before  the  first  figure. 


106  DECIMALS. 

9 

RULE   FOR  DECIMAL  NUMERATION. 

I.  Numerate  from  the  decimal  point,  to  determine  the  de- 
nominator. 

II.  Numerate  towards  the  decimal  point,  to  determine  the 
numerator. 

III.  Read  the  decimal  as  a  whole  number,  giving  it  the 
name  of  its  lowest  decimal  unit,  or  right  hand  figure. 

EXAMPLES   FOR   PRACTICE. 

1.  Write  325  ten-thousandths.  Ans.  .0325. 

2.  Write  four  hundred  ten-thousandths. 

3.  Write  117  ten-thousandths. 

4.  Write  ten  ten-thousandths.  Ans.  .0010. 

5.  Write  250  millionths.  Ans.  .000250. 

6.  Write  twelve  hundred  ten-thousandths. 

7.  Write  9  hundred-thousandths.  Ans.  .00009. 

8.  Read  the  following  decimals. 

.1236        .00061        .32760 
.0080        .720000        040721 

9.  Write  four  hundred,  and  nine  tenths. 

Ans.    400.9. 

10.  Write  twenty-seven,  and  fifty-six  hundredths. 

11.  Write  eighty-five,  and  one  hundred  fifty  thousandths. 

12.  Write  one  thousand,  and  twelve  millionths. 

13.  Write  three  hundred  sixty-five,  and  one  thousand 
eight  hundred  seven  hundred-thousandths. 

Ans.    365.01807. 

14.  Write  nine  hundred  ninety,  and  three  thousand  two 
hundred  fourteen  millionths.  Ans.    990.003214. 

15.  Read  the  following  numbers  : 

71.03       11.0003       34.800000 
126.326     240.01376      9.1263476 


REDUCTION.  •         >    *       107 

REDUCTION. 
CASE  I. 

111.  To  reduce  decimals  to  a  common  denomina- 
tor. 

1.  Reduce  .3,  .09,  .0426,  .214  to  a  common  denominator. 

OPERATION.          ANALYSIS.     A   common   denominator  must 

.3000          contain  as  many  decimal  places  as  is  equal  to 

.0900          the  greatest  number  of  decimal  figures  in  any 

of  the  given  decimals.     We  find  that  the  third 

number  contains  four  decimal  places,  and  hence 

10000  must  be  a  common  denominator.     As  annexing  ciphers 

to  decimals  does  not  alter  their  value,  we  give  to  each  number 

four  decimal  places,  by  annexing  ciphers,  and  thus  reduce  the 

given  decimals  to  a  common  denominator.     Hence, 

RULE.      Giv$  to  qalh  number  the  same  number  of  deci- 
mal places  ^y  annexing  ciphers.  tk      * 

EXAMPLES   FOR   PRACTICE. 

2.  Reduce  .7,  .073,  .42,  .0020  and  .007  to  a  common  de- 
nominator. 

3.  Reduce  .004,  .00032,  .6,  .37  and  .0314  to  a  common 
denominator.  * 

4.  Reduce  1  tenth,  46  hundredths,  15  thousandths,  462 
ten-thousandths,  and  28  hundred-thousandths,  to  a  common 
denominator. 

5.  Reduce  9  thousandths,  9  ten-thousandths,  9  hundred- 
thousandths  and  9  millionths  to  a  common  denominator. 

6.  Reduce  42.07,   102.006,   7.80,  400.01234  to  a  com. 
mon  denominator. 

7.  Reduce  300.3,  8.1003,  14.12614,   210.000009,  and 
1000.02  to  a  common  denominator. 


$'    : 

ia«fc 


QECIMALS. 


CASE   II. 

119.  To  reduce  a  decimal  to  a  common  fraction. 

1.  Reduce  .125  to  an  equivalent  common  fraction. 

OPERATION.  ANALYSIS.  Writing  the  decimal  figures, 
.125  =  TWo-==i  .125,  over  the  common  denominator,  1000, 
we  have  -Jfyjfo$=$. 

RULE.  Omit  the  decimal  point,  supply  the  proper  de- 
nominator, and  then  reduce  the  fraction  toits  lowest  terms. 

EXAMPLES   FOR   PRACTICE. 

1.  Reduce  .08  to  a  common  fraction.  Ans.  •£%. 

2.  Reduce  .625  to  a  common  fraction.  Ans.  f . 

3.  Reduce  .375  to  a  common  fraction.  Ans.  f, 

4.  Reduce  .008  to  a  common  fraction.  Ans.  T^. 

5.  Reduce  .4  to  a  common  fraction.  Ans.  I. 

5 

6.  Reduce  .024  to  a  common  fraction,  Ans.  T4T 


CASE  ill. 

113.  To  rduce  a  common  frffctJrai  to  a  decimal 

2.  Reduce  f  to  its  equivalent  decimal. 

ANALYSIS.  Since  we  can  not  di- 
vide the  numerator  3,  by  4,  we  re- 
duce it  to  tenths  by  annexing  a  ci- 
pher, and  then  dividing  we  obtain  7 
tenths,  and  a  remainder  of  2  tenths. 
Reducing  this  remainder  to  7mn- 
dredths  by  annexing  a  cipher,  and 
dividing  by  4,  we  obtain  5  hun- 
dredths.  The  sum  of  the  quotients 
gives  .75,  the  required  answer. 


OPERATION. 

4)3.0(7  tenths. 
9  Q 

^J.O 

4^20(5  hundredth* 
20 

—      Ans.  .75. 
or  4)3.00 


.75  Ans. 

RULE  I.  Annex  ciphers  to  the  numerator,  and  divide 
~by  the  denominator. 

II.  Point  off  as  many  decimal  places  in  the  result  as  are 
equal  to  the  number  of  ciphers  annexed* 


ADDITION.  109 


EXAMPLES   FOR   PRACTICE. 

1.  Reduce  £  to  a  decimal.  Ans.  .5. 

2.  Reduce  J  to  a  decimal.  Ans.  .25. 

3.  Reduce  f  to  a  decimal.  Ans.  A. 

4.  Reduce  |  to  a  decimal.  Ans.  .8. 

5.  Reduce  $  to  a  decimal.  Ans.  .125. 

6.  Reduce  T95  to  a  decimal.  Ans.  .9. 

7.  Reduce  |  to  to  a  decimal.  Ans.  .625. 

8.  Reduce  ^  to  a  decimal.  Ans.  .04. 

9.  Reduce  T5g  to  a  decimal.  Ans.  .3125. 

10.  What  decimal  is  equivalent  to  £  J  ?  Ans.  .85. 

11.  What  decimal  is  equivalent  to  T3g  ?         ^4ns.  .1875. 

12.  What  decimal  is  equivalent  to  -^1         Ans.  .016. 

ADDITION. 

1  ll.  Since  the  same  law  of  local  value  extends  both  to 
the  right  and  left  of  units'  place;  that  is,  since  decimals  and 
simple  integers  increase  and  decrease  uniformly  hy  the  scale 
of  ten,  it  is  evident  that  decimals  may  be  added,  subtracted, 
multiplied  and  divided  ^n  the  same  manner  as  integers. 

1.  What  is  the  sum  of  4.314,  36.42,  120.0042,  and 
.4276] 

OPERATION.  ANALYSIS.     We  write  the   numbers   so 

4.314  that  the  figures  of  like  orders  of  units  shall 

36.42  stand  in  the  same  columns  ;  that  is,  units 

^  under   units,    tenths  under    tenths,    hun- 

dredths  under  hundredths,  &c.    This  brings 


161  1658  *ne  Decimal  Points  directly  under  each^Dth- 
er.  Commencing  at  the  right  hand,  we  add 
each  column  separately,  and  carry  as  in  whole  numbers,  and  in 
the  result  we  place  a  decimal  point  between  units  and  tenths, 
or  directly  under  the  decimal  point  in  the  numbers  added 
From  this  example  we  derive  the  following 


110  DECIMALS. 

RULE.  I.  Write  the  numbers  so  that  the  decimal  pvint* 
shall  stand  directly  under  each  other. 

II.  Add  as  in  ivlwle  number s}  and  place  the  decimal  point, 
in  the  result,  directly  under  the  points  in  the  numbers  added. 

EXAMPLES   FOR   PRACTICE. 

2.  What  is  the  sum  of  2.7,  30.84,  75.1,  126.414  and 
3.06?  Ans.  238.114. 

3.  What  is  the  sum  of  1.7,  4.45,  6.75,  1.705,  .50    and 
.05?  Ans.  15.155. 

4.  Add  105.7,  19.4,  1119.05,  648.006  and  19.041. 

Ans.  1911.197. 

5.  Add  48.1,  .0481,  4.81,  .00481,  481. 

Ans.  533.96291. 

6.  Add  1.151,  13.29,  116.283,  9.0275  and  .61. 

Ans.  140.3^15. 

7.  Add  .8,  .087,  .626,  .8885  and  .49628. 

8.  What  is  the   sum  of  91.003,    16.4691,  160.00471, 
700.05,  900.0006,  .03^5  ?  Ans.  1867.55891. 

9.  What  is  the  sum  .of  fifty-four,  and  thirty-four  hun- 
dredths;  one,  and\ine   ten-thousanMths ;    thre'e,"  and  two 
hundred  seven  milliomhs ;  twenty-three  thousandths;  eight, 
and  nine  tenths;    four,  and  one  hundred  thirty-five  thou- 
sandths? Ans.  71.399107. 

10.  How  many  acres  of  land  in  four  farms,  containing 
respectively,  61.843   acres,  120.75  acres,  142.4056  acres, 
and  180.750  acres?  Ans.  505.7486. 

1L  How  many  yards  of  cloth  in  3  pieces,  the  first  con- 
taining 21^  yards,  the  second  36|  yards,  and  the  third 
40.15  yards?  Ans.  98.40. 

12.  A  man  owns  4  city  lots,  containing  32|,  36|,  40f, 
42.73  rods  of  land  respectively;  how  many  rods  in  all? 

Ans.  152.205  rods. 


SUBTRACTION. 


Ill 


Ans.  76.9624 


SUBTRACTION. 

115.     From  12 4.2750  take  47.3126. 

OPERATION.  ANALYSIS.     Write  the  subtrahend  un- 

124.2750  der  the  minuend,  placing  units  under 
47.3126  units,  tenths  under  tenths,  &c.  Com- 
mencing at  the  right  hand,  we  subtract 
as  in  whole  numbers,  and  in  the  remain- 
tier  we  place  the  decimal  point  directly  under  those  in  the  num- 
bers above.  If  the  number  of  decimal  places  in  the  minuend 
and  subtrahend  are  not  equal,  they  may  be  reduced  to  the  same 
number  of  decimal  places  before  subtracting,  by  annexing  ci- 
phers. Hence  the 

RULE  1.  Write  the  numbers  so  that  the  decimal  points 
shall  stand  directly  under  each  other. 

II.  Subtract  gs  in  whole  numbers,  and  place  the  decimal 
point  in  the  result  directly  under  the  points  in  the  given 
numbers. 


EXAMPLES  FOR   PRACTICE. 

(2)  (3) 

Minuend,        12.07  37.4562 
Subtrahend,      4.3264  .97 


Remainder,   ^7.7436               36.4862  .628476 

5.  From  463.05  take  17.0613.  Ans.  445.9887. 

6.  From  134.63  take  101.1409.  Ans.  83.4891. 

7.  From  189.6145  take  10.151.  Ans.  179.4635. 

8.  From  671.617  take  116.1.  Ans    555.517. 

9.  From  480.  take  245.0075.  Ans.  234.9925. 

10.  Subtract  .09684  from  .145.  Ans.  .04816. 

11.  Subtract  .2371  from  .2754.  Ans.  .0383. 

12.  Subtract  215.7  from  271.  Ans.  55.3. 

13.  Subtract  .0007  from  107.  Ans.  106.9993. 

14.  Subtract  1.51679  from  27.15.  Ans.  25.63321. 


112  DECIMALS. 

15.  Subtract  37i  from  84.125.  Ans.  46.625. 

16.  Subtract  3|  from  9.3261.  Ans.  5.5761. 

17.  Subtract  25.072  from  112|.  Ans.  87.553. 

18.  A  man  owned  fifty-four  liundredths  of  a  township 
of  land,  and  sold  fifty-four  thousandths  of  the  same,  how 
much  did  he  still  own  1  Ans.  .486. 

19.  From  10  take  three  millionths.     Ans.  9.999997. 

20.  A  man  owning  475  acres  of  land,  sold  at  different 
times  80.75  acres,   100 J   acres,  and  125.625  acres;  how 
much  land  had  he  left  ?     \  Ans.  168.5  acres. 

MULTIPLICATION. 

116.     1.  What  is  the  product  of  .25  multiplied  by  .5. 

OPERATION.  ANALYSIS.     We  first  multiply  as  in  whole 

.25  numbers ;   then,  since  the  multiplicand  has  2 

.5  decimal  places  and  the  multiplier  1,  we  point  off 

~~  2 -{-1=3  decimal  places  in  the  product.     The 

ns>'    reason  for  this  will  be  evident,  by  considering 

both  factors  common  fractions,   and  then  multiplying  as  in 

(99),  thus:    .25=T^an<1.5  =  T5o;   and  ^XiV^Wi 

which  written  decimally  is  .125  Ans.     Hence  the 

RULE.  Multiply  as  in  whole  numbers,  and  from  the 
riff  Jit-  hand  of  the  product  point  off  as  many  figures  for  dec- 
imals as  there  are  decimal  places  in  Loth  factors. 

NOTES.  1.  If  there  be  not  as  many  figures  in  the  product  as  there  are  decimals 
In  both  factors,  supply  the  deficiency  by  prefixing  ciphers. 

2.  To  multiply  a  decimal  by  10,  100, 1000,  &c.,  remove  the  point  as  many  place* 
to  the  right  as  there  are  ciphers  on  the  right  of  the  multiplier. 

EXAMPLES   FOR   PRACTICE. 

(2)  (3)  (4) 

.241  9.4263  .01346 

.7  .5  .06 


.1687  4.71315  .0008076 


MULTIPLICATION.  113 

5.  Multiply  7.1  by  8.2.  Ans.  58.22. 

6.  Multiply  15.5  by  .08.  Ans.  1.24. 

7.  Multiply  8.123  by  .09.  Ans.  .73107. 

8.  Multiply  4.5  by  .15.  Ans.  .675. 

9.  Multiply  450.  by  .02.  Ans.  9. 
10.  Multiply  341.45  by  .007.  Ans.  2.39015. 

•11.  Multiply  3020.  by  .015.  Ans.  45.3ft? 

12.  Multiply  .132  by  .241.  Ans.  .031812. 

13.  Multiply  .23  by  .009.  Ans.  .00207. 

14.  Multiply  7.02  by  5.27.  Ans.  36.9954. 
*»15.  Multiply  .004  by  .04.  Ans.  .00016. 

16.  Multiply  2461.  by  .Q52&__- Ans.  130.1869. 

17.  Multiply  .007853  by  .035^1 — J^s.  .000274855. 

18.  Multiply  25.238  by  12.17.  Ans.  307.14646. 

19.  Multiply  .3272  by  10.  Ans.  3.272. 

20.  Multiply  .3272  by  100.  Ans.  32.72. 

21.  Multiply  .3272  by  1000.  Ans.  327.2. 

22.  Find  the  value  of  .25X«5Xl2.  Ans.  1.5. 

23.  Find  the  value  of  .07x2.4 X-015.      Ans.  00252. 

24.  Find  the  value  of  6JX.8X3.16.        Ans.  16.432. 

25.  If  a  man  travel  3.75  miles  an  hour,  how  far  will  he 
travel  in  9.5  hours'?  Ans.  35.626  miles. 

26.  If  a  sack  of  salt  conialn  94.16  pounds,  how  many 
pounds  will  17  such  sacks  contain  ? 

Ans.  1600.72  pounds. 

27.  If  a  man  spend  .87  of  a  dollar  in  1  day,  how  much 
will  he  spend  in  15.525  days  ? 

Ans.  13.50675  dollars. 

28.  One  rod  is  equal  to  16.5  feet;  how  many  feet  in 
30.005  rods  ?  Ans.  495.0825. 

29.  How  many  gallons  of  molasses  in  .54  of  a  barrel, 
there  being  31.5  gallons  in  1  barrel  ?    A  ns.  17.01  gallons. 


114  DECIMALS. 


DIVISION. 

117.     1.  What  is  the  quotient  of  .225  divided  by  .5  ? 
OPERATION.  ANALYSIS.     We  perform  the  division 

,5).225  the  same  as  in  whole  numbers,  and  the 

only  difficulty  we  meet  with  is  in  point- 
.45  Ans.  jng  og1  the  decimal  places  in  the  quotient. 
To  determine  how  many  places  to  point  off,  we  may  reduce  the 
decimals  to  common  fractions,  thus;  .225=-^^  and  5==-^, 
performing  the  division  as  in  (97),  we  have  T22_5j_i__5^___2^5j 
X  '-0  =  -^Aj ;  and  this  quotient  expressed  decimally,  is  .40. 
Here  we  see  that  the  dividend  contains  as  many  decimal  places 
as  are  contained  in  both  divjsor  and  quotient.  Hence  the  fol- 
lowing 

RULE.  Divide  as  in  whole  numbers,  and  from  the  right 
hand  of  the  quotient  point  off  as  many  places  for  decimals 
as  the  decimal  places  in  the  dividend  exceed  those  m  the 
divisor. 

NOTES.  1.  If  the  number  of  figures  in  the  quotient  be  less  than  the  excess  of 
the  decimal  places  in  the  dividend  over  those  hi  the  divisor,  the  deficiency  must 
be  supplied  by  prefixing  ciphers. 

2.  If  there  be  a  remainder  after  dividing  the  dividend,  annex  ciphers,  and  con- 
tinue the  division  ;  the  ciphers  annexed  are  decimals  of  the  dividend. 

3.  The  dividend  must  always  contain  at  least  as  many  decimal  places  as  th« 
divisor,  before  commencing  the  division. 

4.  In  most  business  transactions,  the  division  is  considered  sufficiently  exact 
when  the  quotient  is  carried  to  4  decimal  places,  unless  great  accuracy  is  required. 

5.  To  divide  by  10,  100,  1000,  &c.,  remove  the  decimal  point  as  many  places  to 
the  left  as  there  are  ciphers  on  the  right  hand  of  the  divisor. 

EXAMPLES   FOR  PRACTICE. 

(2)  (3)  (4)  (5) 

.6).426         .8)3.7624        .05)81.60         .009).00207 

.71.  4.703  1632.    *  ~S 


DIVISION.  115 

(6)          (7)          (8) 
.075).9375(12.5  .288)18.0000(.0625  .0025)15.875(6350, 
75          1728  150 

187          720  87 

150          576  75 

375         1440  125 

375         1440  125 

9.  Divide  44  by  .4.  Ans.  110. 

10.  Divide  15  by  .25.  Ans.  60. 

11.  Divide  .3276  by  .42.  .Ans.  .78. 

12.  Divide  .00288  by  .08.  Ans.  .036. 

13.  Divide  .0992  by  .32.  Ans.  .31. 

14.  Divide  17.6  by  44.  Ans.  .5. 

15.  Divide  .0000021  by  .0007.  Ans.  .003. 

16.  Divide  .56  by  1.12.  Ans.  5. 

17.  Divide  1496.04  by  10.  Ans.  149.604. 

18.  Divide  1196.04  by  100.  Ans.  14.9604. 

19.  Divide  1596.04  by  1000.  Ans.  1.49604. 

20.  Divide  4.96  by  100.  Ans.  .0496. 

21.  Divide  10  by  .1.  Ans.  100. 

22.  Divide  100  by  .2.  Ans.  500. 

23.  If  2.5  acres  produce  34.75  bushels  of  wheat,  how 
much  does  one  acre  produce  ?  Ans.  13.9  bushels. 

24.  If  a  man  travels  21.4  miles  a  day,  how  many  days 
will  he  require  to  travel  461.03  miles? 

25.  If  a  man  build  812.5  rods  of  fence  in  100  days, 
how  many  rods  does  he  build  each  day? 

26.  Paid  131.15  for  61  sheep;  how  much  was  paid  for 
each  ?  Ans.  2.15  dollars. 


116  DECIMALS. 

PROMISCUOUS   EXAMPLES. 

1.  Add  twenty-five   hundredths,  six  hundred   fifty-four 
thousandths,  one  hundred  and  ninety-nine  thousandths,  and 
seven  thousand  five  hundred  sixty-nine  ten-thousandths. 

Ans.  1.8599. 

2.  From  ten  take  ten  thousandths.  Ans.  9.99. 

3.  What  is  the  difference  between  forty  thousand,  and 
forty  thousandths?  Ans.  39999.960. 

4.  Multiply  sixty-five  hundredths,  by  nine  hundredths. 

Ans.  .0585. 

5.  Divide  324  by  6400.  Ans.  .050625. 

6.  Reduce  .125  to  a  common  fraction.  Ans.  |. 

7.  Reduce  J  to  a  decimal  fraction.  Ans.  .875. 

8.  Divide  .016Q04  by  .004.  Ans.  4.001. 

9.  Reduce  JX  to  a  decimal  fraction.  Ans.. 68. 

10.  Reduce"  .4,  .007,  .1142,  .036,  .00015,  and  .42,  to  a 
common  denominator. 

11.  At  13.9  dollars  a  ton,  what  will  2.5  tons  of  hay  cost? 

Ans.  34.75  dollars. 

12.  If  a   pound  of   sugar  cost  .09   dollars,  how  many 
pounds  can  be  bought  for  5.85  dollars?   Ans.  65  pounds. 

13.  If  40.02  bushels  of  potatoes  are  raised  upon  1  acre 
of  land,  how  many  acres  would  be  required  to  raise  4580.64 
bushels?  Ans.  114.458  acres. 

14.  At  11  dollars  a  ton,  how  much  hay  can  be  bought 
for  13.75  dollars?  Ans.  1.25  tons. 

15.  If  a  man  travel  32.445  miles  in  a  day,  how  far  can 
he  travel  in  .625  of  a  day?  Ans.  20.278125  miles. 

16.  If  2  pounds  of  sugar  cost  .1875  dollars,  what  will 
be  the  cost  of  10  pounds?  Ans.  .9375  dollars. 

17.  If  3  barrels  apples  cost  19.125  dollars,  what  will  be 
the  cost  of  100  barrels  ?—  Ans.  337.5  dollars. 


UNITED    STATES    MONEY.  117 


UNITED  STATES  MONEY. 

118.  United   States  Money    is  the  legal  currency 
of  the  United  States,  and  was  established  by  act  of  Con- 
gress August  8,  1786.     Its  denominations  and  their  rela- 
tive "values  are  shown  in  the  following 

TABLE. 

10  mills  (m.)  make  1  cent, c. 

10  cents  "     1  dime, d. 

*  10  dimes  "     1  dollar, $. 

10  dollars  "     1  eagle, E. 

NOTE. — The  currency  of  the  United  States  is  decimal  currency,  and  is  sometimes 
called  federal  Money. 

119.  The  character,  $,  before  any  number  indicates 
that  it  expresses  United  States  money.     Thus  $75  expresses 
75  dollars. 

120.  The    dollar    is    the    unit    of   United    States 
money;  dimes,  cents,  and  mills  are  fractions  of  a  dollar, 
and  are  separated  from  the  dollar  by  the  decimal  point  (.) ; 
thus,  two  dollars  one  dime  two  cents  five  mills  are  written 
$2.125. 

121.  By  examining  the  above  table  we  find 

1st.  That  the  dollar  being  the  unit,  dimes,  cents  and 
mills  are  respectively  tenths,  hundredths  and  thousandths 
of  a  dollar. 

2d.  That  the  denominations  of  United  States  money 
increase  and  decrease  the  same  as  simple  numbers  and  dec- 
imals, and  are  expressed  according  to  the  decimal  system  of 
notation. 

Hence  we  conclude  that 

United  States  money  may  le  added,  subtracted,  multi- 
plied and  divided  in  the  same  manner  as  decimals. 


118  UNITED   STATES   MONEY. 

Dimes  are  not  read  as  dimes,  but  the  two  places  of  dimes 
and  cents  are  appropriated  to  cents;  thus  1  dollar  3  dimes  2 
cents,  or  $1.32,  are  read  one  dollar  thirty-two  cents;  hence, 

When  the  number  of  cents  is  less  than  10,  we  write  a 
cipher  before  it  in  the  place  of  dimes. 

NOTE.    The  half  cent  is  frequently  written  as  5  mills  :  thus,  24%  cents,  written 
$.245. 

EXAMPLES   FOR   PRACTICE. 

1.  Write  five  dollars  twenty-five  cents.        Ans.  85.25. 

2.  Write  four  dollars  eight  cents.  Ans.  $4.08*^ 

3.  Write  twelve  dollars  thirty-six  cents. 

4.  Write  seven  dollars  sixteen  cents. 

6.  Write  ten  dollars  ten  cents. 

7.  Write  sixty-five  cents  four  mills.  $.654. 

8.  Write  one  dollar  five  cents  eight  mills.         $1.058. 

9.  Write  eighty-seven  cents  five  mills.        Ans.  $.875. 

10.  Write  one  hundred  dollars  one  cent  one  mill. 

Ans.  $100.011. 

11.  Read  $4.07;  $3.094;  $10.50;  $25.02. 

KEDUCTION. 
122.     1.  How  many  cents  are  there  in  75  dollars  7 

OPERATION.  ANALYSIS.     Since  in  1  dollar  there  are 

75  100  cents,  in  75  dollars  there  are  75  times 

100  100  cents  or  7500  cents.     To  multiply  by 

10,  100,  &c.,  we  annex  as  many  ciphers  to 

7500  cents.      the  muitipiicand  as  there  are  ciphers  in  the 

multiplier,  (  62  ).     Hence 

To  change  dollars  to  cents,  multiply  by  100 ;  that  is,  an- 
nex  TWO  ciphers.     And 

To  change  dollars  to  mills,  annex  THREE  ciphers. 
To  change  cents  to  mills,  annex  ONE  cipher. 


REDUCTION.  119 

EXAMPLES   FOR   PRACTICE. 

2.  Reduce  $24  to  cents.  Ans.  2400  cents. 

8.  Reduce  $42  to  cents.  Ans.  4200  cents. 

4.  Reduce  $14  to  mills.  Ans.  14000  mills. 

5.  Reduce  $102  to  cents. 

6.  Change  $35  to  mills. 

7.  Change  66  cents  to  mills.  Ans.  660  mills. 

8.  Change  73  cents  to  mills. 

NOTE.    To  change  dollars  and  cents,  or  dollars,  cents,  and  mills  to  mills,  remor* 
the  decimal  point  and  sign,  $. 

9.  Change  $4.28  to  cents.  Ans.  428  cents. 

10.  Change  $18.07  to  cents.  Ans.  1807  cents. 

11.  Change  $6.325  to  mills.  Ans.  6325  mills. 

12.  In  $7.01  how  many  cents? 

13.  In  94  cents  how  many  mills  ? 

14.  In  $51  how  many  cents  1 

1.  In  3427  cents  how  many  dollars? 

OPERATION.  ANALYSIS.     Since  100  cents  equal 

1/00)34/27  1  dollar,  3427  cents  equal  as  many 

• dollars  as   100   is    contained  times 

$34.27  Ans.  m  3427,  which  is  34.27  times. 
To  divide  by  10,  100,  &c.,  cut  off  as  many  figtires  from  the  right 
of  the  dividend  as  there  are  ciphers  in  the  divisor,  (  72  )• 
Hence 

To  change  cents  to  dollars,  divide  by  100 ;  that  is,  point 
off  TWO  figures  from  the  right.    And 

To  change  mills  to  dollars,  point  off, THREE  figures. 
To  change  mills  to  cents,  point  off  ONE  figure. 

EXAMPLES   FOR   PRACTICE. 

2.  Change  972  cents  to  dollars.  Ans.  $9.72. 

3.  Change  1609  cents  to  dollars.  Ans.  $16.09. 

4.  Change  3476  mills  to  dollars  Ans.  $3.476 


120  UNITED   STATES   MONEY. 

5.  In  34671  cents  how  many  dollars  ? 

6.  10307  cents  how  many  dollars  1 

7.  In  203062  mills  how  many  dollars?     Ans.  $203.062. 

8.  Reduce  672  mills  to  cents.  Ans.  $.672. 

9.  Reduce  3104  mills  to  dollars. 
10.  Reduce  17826  cents  to  dollars. 

ADDITION. 

123.     1.  What  is  the  sum  of  $12.50,  $8.125,  $4.076, 
$15.375  and  $22? 

OPBBATION. 
$12.50 

8.125  ANALYSIS.   "Writing  dollars  under  do' - 

4.076  lars,  cents  under  cents,  &c.,  so  that  the 

15.375  decimal  points  shall  stand  under  each 

22.000  other,  we  add  and  point  off  as  in  ad- 

dition of  decimals.    Hence  the  following 
$oZ.U7o  Ans. 

RULE.  I.  Write  dollars  under  dollars,  cents  under  cents,  &c. 
II.  Add  as  in  simple  numbers,  and  place  the  point  in  the 
amount  as  in  addition  of  decimals. 

EXAMPLES   FOR   PRACTICE. 

(2)  (3)  (4)  (5) 

$  42.64  $100.375  $750.00  $1042.875 

126.085  13.09  140.07  427.035 

304.127  65.82  35.178  50.50 

14.42  400.00  6.004  7.08 


6.  What  is  the  sum  of  30  dollars  9  cents ;  200  dollars  63 
cents ;  27  dollars  36  cents  4  mills,  and  10  dollars  16  cents  ? 

Ans.  $268.244. 

7.  Add  390  dollars  37  cents  5  mills,  187  dollars  50 
cents,  90  dollars  5  cents  5  mills,  and  400  dollars  40  cents. 

Ans.  $1068.33. 


ADDITION.  121 

,  8.  A  lady  paid  $45.40  for  some  furs,  $12.375  for  a  dress, 
$5  for  a  bonnet  and  $1.125  for  a  pair  of  gloves;  how  much 
did  she  pay  for  all  ? 

9.  A  farmer  sold  a  cow  for  $20,  a  horse  for  $96.50,  a 
yoke  of  oxen  for  $66.875,  and  a  ton  of  hay  for  $9.40; 
how  much  did  he  receive  for  all  ?  Ans.  $192.775. 

10.  Bought  a  hat  for  $4.50,  a  pair  of  boots  for  $5.62  4, 
an  umbrella  for  $2.12^,  and  a  pair  of  gloves  for  $.87^ ; 
wfeat  was  the  cost  of  the  whole?  Ans.  $13.125. 

11.  A  grocer  bought  a  barrel  of  sugar  for  $17.84,  a  box 
of  tea  for  $36.12±,  a  cheese  for  $4,  and  a  tub  of  butter  for 
$7.09;  what  was  the  cost  of  all  ? 

12.  A  merchant  bought  a  quantity  of  goods  for  $458.25, 
paid  for  duties  $45;  for  freights  $98.624,  and  for  insur- 
ance $16.40;  how  much  was  the  whole  cost? 

Ans.  $618.275. 

13.  Bought  some  sugar  for  $1.75,  some  tea  for  $.90, 
some  butter  for  $2.12^,  some  eggs  for  $.37|,  and  some  spice 
for  $.25 ;  what  was  the  cost  of  the  whole  ?     Ans.  $5.40. 

14.  Paid  for  building  a  house  $1045.75,  for  painting  the 
same   $275.60,  for   furniture   $648.87|,   and   for   carpets 
$105.10;  what  was  the  cost  of  the  house  and  furnishing? 

Ans.  $2075.325. 

15.  A  farmer  receives  120  dollars  45  cents  for  wheat, 
36  dollars  624  cents  for  corn,  14  dollars  9  cents  for  pota- 
toes, and  63  dollars  for  oats ;  how  much  does  he  receive  foi 
the  whole] 

16.  A  lady  who  went  shopping,  bought  a  dress  for  7  dol- 
lars 27  cents,  trimmings  for  874  cents,  some  tape  for  6  cents, 
some  thread  for  12^  cents,  and  some  needles  for  9  cents; 
how  much  did  she  pay  for  all  ?  Ans.  $8.42. 

6 


122  UNITEI )   STATES   MONEY. 

SUBTRACTION. 

124.     1.  From  246  dollars  82  cents  5  mills,  take  175 
dollars  27  cents. 

OPERATION.  ANALYSIS.     Writing  the  less  num- 

$246.825  ber  under  the  greater,  dollars  under 

175.27  dollars,   cents  under   cents,    &c.,    we 

subtract  and  point  off  in  the  result  as 

$71.555  Ans.         jn   subtraction  of  decimals.      Hence 

RULE.     I.    Write  the   subtrahend  under   the   minuend, 

dollars  under  dollars,  cents   under  cents,  &G., 

II.  Subtract  as  in  simple  numbers,  and  place  the  point  in 
the  remainder  as  in  subtraction  of  decimals. 

EXAMPLES   FOB,   PEACTICE. 

(2)  (3)  (4)  (5) 

From  $125.05        $327.105        $112.000         $43.375 
Take       43.278        100.09  .875  2.06 

Ans.  $81.772       $227.015      $111.125         $41.315 

6.  From  $3472.50  take  $1042.125.  Ans.  $2430.375. 

7.  From  $540  take  $256.67.  Ans.  $283.33. 

8.  From  $82.04  take  $80.625.  Ans.  $1.415. 

9.  From  3  dollars  10  cents,  take  75  cents.^4rcs.$2.35. 

10.  From  10  dollars,  take  5  dollars  10  cts.  Ans.  $4.90. 

11.  From  100  dollars,  take  50  dollars  50  cents. 

12.  From  1001  dollars  9  cents,  take  300  dollars. 

13.  From  2  dollars,  take  75  cents.  Ans.  $1.25. 

14.  From  96  cents,  take  12J  cents.  Ans.  $.835. 

15.  From  1  dollar  take  25  cents.  Ans.  $.75. 

16.  From  50  cents  take  37  cents  5  mills.   Ans.  $.125. 

17.  From  5  dollars,  take  50  cents  8  mills.  A ns. $4.492. 

18.  From  4  dollars,  take  J  dollar  40  cents  5  mills. 

19.  Sold  a  horse  for  $200,  which  was  $45.50  more  than 
he  cost  me;  hcrw  much  did  he  cost  me  ?     Ans.  $154.50 


SUBTRACTION.  1*28 

20.  A  man  bought  a  farm  for  $4640,  and  sold  it  for 
$5027.50 ;  how  much  did  he  gain  ?  Ans.  $387.50. 

2^.  Borrowed  $25  and  returned  $15.60 ;  how  much  re- 
mained unpaid  ?  Ans.  9.40. 

22.  A  merchant  having  $10475,  paid  $2426  for  a  store, 
and  $5327.875  for  goods;  how  much  money  had  he  left1? 

Ans.  $2721.125. 

23.  Bought   a   sack   of  flour   for   $3.12^ ;    how   much 
change  must  I  receive  for  a  5  dollar  bill  ?     Ans.  $1.875. 

24.  Bought  groceries  to  the  amount  of   $1.875 ;   how 
much  change  must  I  receive  for  a  2  dollar  bill  ? 

Ans.  12^  cents. 

25.  Paid; $3 7 5  for  a  pair  of  horses,  and  sold  one  of  them 
for  $215.50};  how  much  did  the  other  one  cost  me  ? 

Ans.  $159.50. 

26.  I  started  on  a  journey  with  $50  and  paid  $10.62£ 
railroad  far$,  $7.38  stage  fare,  $5.96  for  board  and  lodging, 
and  $.75  fo£ porterage;  how  much  money  had  I  left  V 

Ans.  $25.285. 

27.  A  faflner  sold  some  wool  for  $27.16,  and  a  ton  of  hay 
for  $14.80.  'He  received  in  payment  a  barrel  of  flour  worth 
$6.875,  and  Qie  remainder  in  money  ;  how  much  money  did 
he  receive?  '  Ans.  $35.085. 

28.  A  woman  sold  a  grocer  some  butter  for  $1.48,  and 
some  eggs  for  $.94.     She  received  a  gallon  of  molasses  worth 
40  cents,  a  pound  of  tea  worth  75  cents,  and  a  pound  of 
starch  worth  124  cents ;  how  much  is  still  her  due  1 

Ans.  $1.145. 

29.  A  tailor  bought  a  piece  of  broadcloth  for  $87.50, 
and  a  piece  of  cassimere  for  $62.75.     He  sold  both  pieces 
for  $170.87^;  how  much  did  he  gain  on  both1? 

Ans.  $20.625. 


124  UNITED  STATES   MONEY. 

MULTIPLICATION. 
125.     1.  Multiply  $26.145  by  34. 

OPERATION. 

ANALYSIS.     We  multiply  as  in  sim- 
ple numbers,   always  regarding  the 
104580  multiplier  as  an  abstract  number,  and 

78435  point  off  from  the  right  hand  of  the 

result,  as  in  multiplication  of  decimals. 


$888.930  Ans.        Hence  the  following 
RULE.     Multiply  as  in  simple  numbers,  and  place  the 
point  in  the  product  as  in  multiplication  of  decimals. 

EXAMPLES   FOR   PRACTICE.         | 

(2)  (3)  (4)  j(5) 

$327.48         $82.375        $160.09         $$7.875 
15  46  {      87  123 

6.  What  cost  8  cords  of  wood,  at  $3.50  ?    fens.  $28. 

7.  What  cost  14  barrels  of  flejur,  at  $5.85  Sbarrel  ? 

8.  What  cost  25  bushels  of  cforn,  at  75  cems  a  bushel  ? 

9.  4t  $2.125  a  yard,  what  wfll  18  yards  otjsilk  cost? 

10.  At  $.8?5: apiece,  what  will  be  the  cost  o¥  9  turkeys? 

11.  A  farmer  sold  40  bushels  .of  potatoes  &  37£  cents  a 
bushel,  and  2l  barrels  of  apples  at  $2.25  ^barrel;  how 
much  did  he'r'eWive  for  both  ?  Ans.  $62.25. 


11.  Bought  124>*apres  of  land  at  $35.75  an  acre,  and 
sold  the  whole  for  $6iOQp';  did  I  gain  or  lose,  and  how 
much?  Ans.  $1567. 

13.  What  will  be  the  cost  of  275  bushels  of  oats,  at  42 
cents  a  bushel  ?  Ans.  $115.50. 

14.  A  grocer  bought  160  pounds  of  butter,  at  14  cents 
a  pound,  and  paid  25  pounds  of  tea,  worth  56  cents  a  pound, 
and  the  remainder  in  cash;  how  much  money  did  he  pay? 


DIVISION.  125 

15.  What  will  be  the  cost  of  15  yards  of  broadcloth,  at 
$4.87 £  a  yard]  Ans.  $73.125. 

16.  A    grocer   bought  a  tub  of  butter  containing   84 
pounds,   at  12 i  cents  a  pound,  and  sold  the  same  at  15 
cents  a  pound ;  how  much  did  he  gain  ?         Ans.  $2.10. 

17.  A  farmer  took  3  tons  of  hay  to  market,  for  which 
he  received  $9.38  a  ton.     He  bought  2  barrels  of  flour,  at 
$6.94  a  barrel,  and  12  pounds  of  tea,  at  $.625  a  pound ; 
how  much  money  had  he  left  ? 

Ans.  $6.76. 

DIVISION. 
126.     1.  Divide  $136  by  64. 

64)$136.000($2.125  Ans. 
128 

~~80 

64 

ANALYSIS.     We  divide  as  in 

~,QQ  simple  numbers,  and  as  there  is 

i  og  a  remainder  after  dividing  the 

dollars,  we  reduce  the  dividend 

320  to  mills,  by  annexing  three  ci- 
320  phers,  and  continue  the  divis- 
ion.  Hence  the  following 

KULE.     Divide  as  in  simple  numbers,  and  place  the  point 
in  the  quotient,  as  in  division  of  decimals. 

NOTE.    1.  In  business  transactions  it  is  never  necessary  to  carry  the  division 
further  than  to  mills  in  the  quotient. 

EXAMPLES   FOR   PRACTICE. 

(2)  (3)  (4)  (5) 

5)$43.50     10)$36.00     8)$371.          12)$169.50 


$8.70  $3.60          $46.375  $14.125 


126  UNITED    STATES   MONET. 

6.  Divide  $13.75  by  11.  Ans.  $1.25. 

7.  Divide  $162.  by  36.  Ans.  $4.50. 

8.  Divide  $246.30  by  15.  Ans.  $16.42. 

9.  Divide  $1305.  by  18.  Ans.  $72.50. 

10.  Divide  $2.25  by  9.  Ans.  $.25. 

11.  Divide  $658  by  280.  Ans.  $2.35. 

12.  Divide  $195.75  by  29.  Ans.  $6.75. 
13    Divide  1388  by  100.  Ans.  $13.88. 

14.  Divide  $2675.75  by  278.  Ans.  $9.625. 

15.  Divide  $68  by  32.  Ans.  $2.125. 

16.  Paid  $168.48  for  144  bushels  of  wheat;  what  waa 
the  price  per  bushel  ?  Ans.  $1.17. 

17.  Paid  $2.80  for  35  pounds  of  sugar;  what  was  the 
price  per  pound  ?  Ans.  $.08. 

18.  If  54  cords  of  wood  cost  $135,  what  is  the  price  per 
cord?  Ans.  $2.50. 

19.  Bought  125  bushels  of  oats  for  $62.50 ;  what  was 
the  cost  per  bushel  ?  Ans.  $.50. 

20.  If  70  barrels  of  apples  cost  $175,  how  much  will  1 
barrel  cost?  Ans.  $2.50. 

21.  If  100  acres  of  land  cost  $3156.50,  how  much  will 
be  the  cost  of  1  acre]  Ans.  $31.565. 

22.  Paid  $148.75  for  170  bushels  of  barley;  how  much 
was  the  cost  per  bushel  ?  Ans.  $.875. 

23.  If  13  pounds  of  tea  cost  $9.88,  how  much  will  1 
pound  cost  ? 

24.  Bought  2500  pounds  of  butter  for  $625 ;  how  much 
was  the  cost  per  pound  ?  Ans.  25  cents. 

25  Bought  2450  pounds  of  pork  for  $153.12^;  how 
much  was  the  cost  per  pound  1  Ans.  6|  cents. 

26.  Bought  4  barrels  of  sugar,  each  containing  200 
pounds,  for  $72 ;  what  was  the  cost  per  pound  ? 


PROMISCUOUS   EXAMPLES.  127 

PROMISCUOUS    EXAMPLES. 

1.  A  merchant  bought  14  boxes  of  tea  for  $560  ;  but  it 
being  damaged,  he  was  obliged  to  sell  it  for  $106.75  less 
than  he  gave  for-  it ;  how  much  did  he  receive  a  box  ? 

Ans.  $32.375. 

2.  A  farmer  sold  120  bushels  of  wheat,  at  $1.12^  a 
nushel,  and  received  in  payment  27  barrels  of  flour;  what 
did  the  flour  cost  him  per  barrel  ? 

3.  If  35  yards  of  cloth  cost  $122.50,  how  much  will  29 
yards  cost?  Ans.  $101.50. 

4.  If  4  tons  of  coal  cost  $35.50,  how  much  will  12  tons 
cost?  Ans.  $106.50. 

5.  If  29  pounds  of  sugar  cost  $3.625,  how  much  will  15 
pounds  cosf?  Ans.  $1.875. 

6.  If  12  barrels  of  flour  cost  $108,  how  much  will  18 
barrels  cost  ?  Ans.  $162. 

7.  If  3  bushels  of  wheat  cost  $4.35,  how  much  will  30 
bushels  cost  ?  Ans.  $43.50. 

8.  A   man   bought   a   farm   containing   125   acres,   for 
$2922.50 ;  for  how  much  must  he  sell  it  per  acre  to  gain 
$500  ?  Ans.  $27.38. 

9.  A  farmer  exchanged  50  bushels  of  corn  worth  70 
cents  a  bushel,  for  28  bushels  of  wheat;  how  much  was  the 
wheat  worth  a  bushel.  Ans.  $1.25. 

10.  A  person  having  $15000,  bought  30  hales  of  cotton 
each  bale  containing  940  pounds,  at  10  cents  a  pound ;  he 
next  paid  $6680  for  a  house,  and  then  bought  1000  barrels 
of  flour  with  what  money  he  had  left ;  how  much  did  the 
flour  cost  him  per  barrel  ?  Ans.  $5.50. 

NOTE.  For  a  full  and  complete  development  and  application  of  Decimals  and 
United  Statea  money,  the  pupil  is  referred  to  the  A.uthor's  Progressive  Practical 
and  Higher  Arithmetic. 


128  UNITED    STATES   MONEY. 

BILLS. 

127.  A  Bill,  in  business  transactions,  is  a  written  state- 
ment of  articles  bought  or  sold,  together  with  the  prices  of 
each,  and  the  whole  cost. 

Find  the  cost  of  the  several  articles,  and  the  amount  or 
footing  of  the  following  bills : 

ao 

CHICAGO,  Sept.  20, 1861. 
MR.  J.  C.  SMITH, 

JBo't.  of  SILAS  JOHNSON, 

36  pounds  sugar  at  8  cents  a  pound,  $2.88 

18  pounds  coffee  at  15  cents  a  pound,  2.70 

24  pounds  butter  at  18  cents  a  pound,  4.32 
10  dozen  eggs  at  12£  cents  a  dozen,  1.25 

4  gallons  molasses  at  44  cents  a  gallon,  1.76 

Ans.  $12.91. 

(20 

ROCHESTER,  Jan.  25, 1862. 
JOHN  DABNEY,  ESQ., 

Bo't.  of  BARDWELL  &  Co., 
14  pounds  coffee  sugar  at  11  cents  a  pound,      $1.54 

6  pounds  Y.  H.  tea  at  62 £  cents  a  pound,         3.75 

25  pounds  No.  1  mackerel  at  6  cents  a  pound,    1.50 

5  bushels  potatoes  at  37  J  cents  a  bushel,         1.875 
3  gallons  syrup  at  80  cents  a  gallon,  2.40 

7  dozen  eggs  at  16  cents  a  dozen,  1.12 

Received  Payment,  Ans.  $12.185 

Bardwell  &  Co., 

per  Adams 


BILLS.  129 

(3.) 

MEMPHIS,  Aug.  20, 1862 
Mr.  S.  P.  HAILE, 

JBo't  of  PATTERSON  &  Co., 
20  chests  Green  Tea  at  $22.50 
16     «      Black      «   at    18.75 

14  "       Imperial  «   at     32.87£ 

15  sacks  Java  Coffee  at     17.38 
25  boxes  Oranges        at      4.62| 

Received  payment,  $1586.575. 

Patterson  &  Co., 

(40 

OSWEGO,  Sept.  4, 1861. 

JAMES  COROVAL  &  Co., 

Bd*t.  of  COLLINS  &  SON, 

12  yards  Broadcloth  at  $3.84 

18     "       Cassimere  "•  2.25 

10     "        Satinet  "  .87£ 

42     "       Flannel  "  .45 

35    "       Black  Silk  "  1.18 

$155.53. 
(5.) 

BOSTON,  April  10, 1862. 
J.  GK  BENNET  &  SON, 

Bc?t.  of  BUTLER,  KINO  &  Co., 
14  Plows        at     $10.50 
8  Harrows     "          9.80 
120  Shovels      "  .90 

175  Hoes          «  .62' 

$442.775. 


130  COMPOUND  LUMBERS. 

COMPOUND  NUMBERS. 

128.  A  Simple  Number  is  either  an  abstract  number, 
or  a  concrete  number  of  but  one  denomination.     Thus,  48, 
926;    48  dollars,  926  miles. 

129.  A  Compound  Number  is  a   concrete   number 
whose  value  is  expressed  in  two  or  more  differfij^Renomi- 
nations.     Thus,  32  dollars  15  cents ;  15  days  4  hours  25 
minutes. 

130.  A  Scale  is  a  series  of  numbers,  descending  or  as 
cending,  used  in  operations  upon  numbers. 

NOTE.    In  simple  numbers  and  decimals  the  scale  is  uniformly  10;  in  compound 
numbers  the  scales  are  varying. 

CURRENCY. 

I.  UNITED  STATES  MONEY. 

131.  The  currency  of  the  United  States  is  decimal  cur- 
rency, and  is  sometimes  called  Federal  Money. 

TABLE. 

10  mills  (m.)  make  1  cent, ct 

10  cents  *'      1  dime,  . . . .  d. 

10  dimes  **      1  dollar, $. 

10  dollars          "      1  eagle, B. 

UNIT  EQUIVALENTS. 

ct.  m. 

d.  1—10 

$  1—10—100 

B          1  —     10  —    100  —     1000 

1  —  10  —  100  —  1000  —  10000 

SCALE — uniformly  10. 

COINS.     The  gold  coins  are  the  double  eagle,  eagle,  halt 
eagle,  quarter  eagle,  three-dollar  piece  and  dollar. 

The  silver  coins  are  the  half  and  quarter  dollar,  dime  and 
half  dime,  and  three-cent  piece. 
nickel  coin  is  the  cent. 


MONEY   AND    CURRENCIES.  131 

II.    CANADA   MONEY. 

132.  The  currency  of  the  Canadian  provinces  is  deci- 
mal, and  the  table  and  denominations  are  the  same  as  those 
of  the  United  States  money. 

NOTE  The  decimal  currency  was  adopted  by  the  Canadian  Parliament  in  1868, 
and  the  Act  took  effect  in  1859.  Previous  to  the  latter  year  the  money  of  Canada 
was  reckoned  in  pounds,  shillings,  and  pence,  the  same  as  in  England. 

COINS.     The  new  Canadian  coins  are  silver  and  copper. 
The  silver  coins  are  the   shilling   or  20-cent  piece,  the 
dime,  and  half  dime. 

The  copper  coin  is  the  cent. 

NOTB.  The  20-cent  piece  represents  the  value  of  the  shilling  of  the  old  Cana- 
da Currency. 

III.    ENGLISH    MONEY. 

133.  English  or  Sterling  money  is  the  currency  of 
Great  Britain. 

TABLE. 

4  farthings  (far.  or  qr.)  make  1  penny, , d. 

12  pence  "      1  shilling, 8. 

20  shillings  "      1  pound  or  sovereign.. .  .£  or  sov. 

UNIT    EQUIVALENTS. 

d.  far. 

..  1  =      4 

£,  or  SOT.  1    =       12    =       48 

1  =  20  =  240  =  960 
SCALE — ascending,  4,  12,  20 ;  descending,  20,  12,  4. 

NOTH.  Farthings  are  generally  expressed  as  fractions  of  a  penny  ;  thus,  1  far., 
sometimes  called  1  quarter,  (qr.)  =}£d.;  3  far.=%d. 

Coixs.  The  gold  coins  are  the  sovereign  (=  £1)  and  the 
half  sovereign,  (=  10s.) 

The  silver  coins  are  the  crown  (=  5s.),  the  half  crown, 
(=  2s.  6d.),  Ijie  shilling,  and  the  6-penny  piece. 

The  copper  coin*  are  the  penny,  half-penny,  and  farthing. 


132  COMPOUND   NUMBERS. 

WEIGHTS. 

1 34.  Weight  is  a  measure  of  the  quantity  of  matter  a 
body  contains,  determined  according  to  some  fixed  standard. 

I.   TROY  WEIGHT. 

135.  Troy  Weight  is  used  in  weighing  gold,  silver, 
and  jewels;  in  philosophical  experiments,  &c. 

TABLE. 
24  grains  (gr.)  make  1  pennyweight, . . .  pwt.  or  dwt. 

20  pennyweights  "    1  ounce, oz. 

12  ounces  "    1  pound, Ib. 

UNIT  EQUIVALENTS. 

pwt.  gr. 

oz.  1—24 

ib.         1—20—480 
1  —  12  —  240  —  5T60 
SCALE— ascending,  24,  20,  12  ;  descending,  12,  20,  24. 

II.  AVOIRDUPOIS  WEIGHT. 

1 36.  Avoirdupois  Weight  is  used  for  all  the  ordinary 
purposes  of  weighing. 

TABLE. 

16  drams  (dr.)        make  1  ounce, oz. 

16  ounces  "       1  pound, Ib. 

100  Ib.  "       1  hundred  weight,  .cwt 

20  cwt.,  =  2000    Ibs.,      1  ton, T. 

UNIT  EQUIVALENTS. 

or,.  dr. 

«»•  1  -  16 

cwt  1  —        16  —        256 

T.  —     1  —     100  —    1600  —    25600 

1  —  20  —  2000  —  32000  —  512000 

SCALE— ascending,  16,  16,  100,  20;  descending,  20,  100,  10, 

ia 


WEIGHTS.  133 

NOTE.  The  long  or  gross  ton,  hundred  weight,  and  quarter  were  formerly  in  com- 
mon use  ;  but  they  are  now  seldom  used  except  in  estimating  English  goods  at  the 
U  8.  custom-house,  and  in  freighting  and  wholesaling  coal  from  the  Pennsylvania 
mines. 

LONG   TON    TABLE. 

28  lb.  make  1  quarter,  qr. 

4  qr.    —  112  lb.        "      1  hundred  weight,  cwt. 

20  cwt.  —  2240  lb.     "      1  ton,  T. 

The  following  denominations  are  also  in  use: 

56  pounds  make  1  firkin  of  butter. 
196       "  "     1  barrel  of  flour. 

200       "          "     1       "      "  beef,  pork,  or  fish. 
280      "          "     1  bushel,  "  salt  at  the  N.  Y.  State  salt  works 

32      "          "     1       "      *  oats. 

48       "          «     1      "      "  barley. 

56      "          "     1       "      "  corn  or  rye. 

60      "          "     1       "      "  wheat. 

III.     APOTHECARIES'    WEIGHT. 

137.  Apothecaries'  Weight  is  used  by  apothecaries 
and  physicians  in  compounding  medicines ;  but  medicines 
are  bought  and  sold  by  avoirdupois  weight. 

TABLE. 

20  grains  [gr.]  make  1  scruple sc.  or  3  • 

3  scruples  "  1  dram, dr.  or  3  . 

8  drams  "  1  ounce, oz.  or  §. 

12  ounces  "      1  pound lb.  or  ft> 

UNIT   EQUIVALENTS. 

sc.  grr. 

d.  1—20 

oz.  1  —       3  —       60 

n>.          1          8  —    24  —    480 
1  —  12  —  96  —  288  —  5760 

SCALE — ascending,  20,  3,  8,  12;  descending,  12,8,8, 
20 


134  COMPOUND    NUMBERS. 

138.  COMPARATIVE   TABLE   OP   WEIGHTS. 

Troy.  Avoirdupois.  Apothecaries. 

1  pound  —  5760  grains,     =  7000  grains,  —  5760  grains, 
1  ounce    —    480      »«          —  437.5     "        —     480     « 

175  pounds,  —  144  pounds,  —     175  pounds. 

MEASURES   OF  EXTENSION. 

139.  Extension  has  three  dimensions — length,  breadth, 
and  thickness. 

A  Line  has  only  one  dimension — length. 

A  Surface  or  Area  has  two  dimensions — length  and 
breadth. 

A  Solid  or  Body  has  three  dimensions — length,  breadth, 
and  thickness. 

I.  LONG    MEASURE. 

140.  Long  Measure,  also  called  Lineal  Measure,  is 
used  in  measuring  lines  or  distances. 

TABLE. 
12    inches  (in.)  make  1  foot, ft 

3     feet  "      lyard, yd. 

5 \  yd.,  or  16£  ft,  "      1  rod, rd. 

40  rods  "      1  furlong, fur. 

8  furlongs,  or  320  rd.,         "      1  statute  mile,. ml 

UNIT  EQUIVALENTS. 

ft.  in. 

yd.  1     -         12 

M.  1-3-36 

tagm  1  —         5J  —       16£  —       198 

ml.       1  —     40  —     220     —     660"  —     7920 
1  _.  8  —  320  —  1760     =-  5280     —  63360 
SCALE— ascending,  12  3,  5',  40,  8;  descending,  8,  40,  5J,  8. 
12. 


MEASURES    OF  EXTENSION. 


185 


The  following  denominations  are  also  in  use  :  — 

j  used  by  shoemakers  in  meas- 
barleycorns  make  1  inch,  j  uring  £he  length  of  thefoot 

(  used  in  measuring  the  height 
inches  "     1  hand,  •<  of  horses  directly  over  the 

(  fore  feet. 
"  "     Ispan. 

"     1  sacred  cubic. 


9          "  " 

21.888   "  " 

3       feet  " 

6  .« 

1.15   statute  miles" 


1  pace. 


1  fathom, 


geographic 


or 


measurinS  Depths 

1  league. 

)  ,   ,  j  of  latitude  on  a  meridian  or 

j  L  J  \  of  longitude  on  the  equator. 


8 

60  '* 

69.16   statute 

360      degrees 

NOTES.  1.  For  the  purpose  of  measuring  cloth  and  other  goods  sold  by  the  yard, 
the  yard  is  divided  into  halves,  quarters,  fourths,  eighths,  and  sixteenths.  The  old 
table  of  cloth  measure  is  practically  obsolete. 


**    the  circumference  of  the  earth. 


SURVEYORS'  LONG  MEASURE. 
A  Gunter's  Chain,  used  by  land  surveyors,  is  4 
rods  or  66  feet  long,  and  consists  of  100  links. 

TABLE. 

7.92  inches  (in.)  make  1  link, 1. 

25  links  "  1  rod, rd. 

4  rods,  or  66  feet,  "  1  chain,... ch. 
80  chains  "  1  mile,.... mi. 


UNIT  EQUIVALENTS. 

1.  In. 

rd.  1    •         7.92 

ch.  1  —      25  —      198 

„!.         1  —       4  =-     100  —       792 

1  _  80  —  320  —  8000  —  63360 
SCALE—  ascending,  7.92,  25,  4,  80  ;  descending,  80,  4,  25,  7.92. 

NOTK.    The  denomination,  rods,  is  seldom  used  in  chain  measure,  distances 
bei  ug  taken  In  chains  and  links. 


136 


COMPOUND    NUMBERS. 


II.    SQUARE  MEASURE. 

143.  A  Square  is  a  figure  having  four  equal  sides,  and 
four  equal  angles  or  corners. 

i  yd. =3  ft.  i  square  yard  is  a  figure  hav- 

ing four  sides  of  1  yard  or  3  feet 
43     each,  as  shown  in  the  diagram. 
Its  contents  are  3X3=9  square 
feet.     Hence 

Thus  a  square  foot  is  12  inches 
i  yd. =s  ft.  long  and  12  inches  wide,  and  the 

contents  are  12x12=144  square  inches.  A  surface  20 
feet  long  and  10  feet  wide,  is  a  rectangle,  containing  20  X 
10=200  square  feet. 

The  contents  or  area  of  a  square,  or  of  any  other  figure 
having  a  uniform  length  and  a  uniform  breadth,  is  found, 
by  multiplying  the  length  by  the  breadth. 

144.  Square  Measure  is  used  in  computing  areas  or 
surfaces ;  as  of  land,  boards,  painting,  plastering,  paving, 
&c. 

TABLE. 


144    square  inches  (sq.  in.)  make  1  square  foot, 

sq.  ft 

9     square  feet                       "       1  square  yard, 

sq.  yd. 

30J  square  yards                    "       1  square  rod, 

sq.  rd. 

40    square  rods                     "      1  rood, 

R. 

4    roods                                 "      1  acre, 

A. 

640    acres                                "      1  square  mile, 

sq.  mi, 

UNIT  EQUIVALENTS. 

«q.  ft. 

nq.  in. 

sq,yd.                    1  — 

144 

Bq.  rd.                 1—                  9  — 

1276 

R.                     1—           30}—          272^— 

30204 

A.          1—           40—       1210—       10890— 

15681  CO 

-  mi    1—       4—         160—       4840—       435  fiO— 

G272640 

1—640— 256dO— 102400— 3097GOO— 27878400— 4014480GOOO 


MEASURES  OF  EXTENSION.  137 

Artificers  estimate  their  work  as  follows  : 

By  the  square  foot :  glazing  and  stone-cutting. 

By  the  square  yard  :  painting,  plastering,  paving,  ceiling, 
and  paper-hanging. 

By  the  square  of  100  feet :  flooring,  partitioning,  roofing, 
slating,  and  tiling. 

Brick-laying  is  estimated  by  the  thousand  bricks;  also  by 
the  square  yard,  and  the  square  of  100  feet. 

NOTES.  1.  In  estimating  the  painting  of  moldings,  cornices,  etc.,  the  measuring 
line  is  carried  into  all  the  moldings  and  cornices. 

2.  In  estimating  brick-laying  by  the  square  yard  or  the  square  of  100  feet,  the 
work  is  understood  to  be  1>£  bricks,  or  12  inches,  thick. 

SURVEYORS'  SQUARE  MEASURE. 

14L5.  This  measure  is  used  by  surveyors  in  computing 
the  area  or  contents  of  land. 

TABLE. 

625  square  links  (sq.  1.)  make  1  pole, P. 

16  poles  "      1  square  chain,  ..sq.  ch. 

10  square  chains  **      1  acre, A. 

640  acres  "       1  square  mile, . . .  sq.  mi. 

36  square  miles  (6  miles  square)  "      1  Township, Tp. 

UNIT  EQUIVALENTS. 

P.  sq.  1. 

eq.  ch.         1  —  625 

A.         1  —      16  —  1000 

Bq.mi.       1  «.     10  —      160  —  10000 

Tp.         1  —       640  —      6400  —     102500  —      64000000 
1  —  36  —  23040  —  230400  —  3686400  —  2304000000 
SCALE — ascending,  625,  16, 10,  630,  36 ;  descending,  36,  640, 
10, 16,  625. 

NOTKS.    1.  A  square  mile  of  land  is  also  called  a  section. 

2.  Canal  and  railroad  engineers  commonly  use  an  engineers^  chain,  which  con- 
•ista  of  100  links,  each  1  foot  long. 


188 


COMPOUND  NUMBERS. 


III.    CUBIC  MEASURE. 

14LO.  A  Cube  is  a  solid,  or  body,  having  six  equal 
square  sides  or  faces. 

If  each  side  of  a  cube  be  1  yard, 
or  3  feet,  1  foot  in  thickness  of 
this  cube  will  contain  3x3x1:= 
9  cubic  feet ;  and  the  whole  cube 
will  contain  3x3X3=27  cubic 
[  ,  I,  '|U^^  feet. 

3  ft.—i  yd.  A  solid,  or  body,  may  have  the 

three  dimensions  all  alike,  or  all  different.  A  body  4  ft. 
long,  3  ft.  wide,  and  2  ft.  thick  contains  4x3x2=24  cu- 
bic or  solid  feet.  Hence  we  see  that 

The  cubic  or  solid  contents  of  a  ~body  are  found  by  multi- 
plying the  length,  breadth,  and  thickness  together. 

147.  Cubic  Measure,  also  called  Solid  Measure,  is 
used  in  estimating  the  contents  of  solids,  or  bodies ;  as 
timber,  wood,  stone,  &c. 


1728 

27 

40 

50 

16 

8 

128 


TABLE. 

make  1  cubit  foot,  . .  cu.  ft. 
u      1  cubic  yard,  cu.  yd. 

1  ton  or  load,  ...T. 
1  cord  foot,  . . .  cd.  ft. 
1  cord  of  wood,  .  Cd. 

(  perch    of  ) 
1  -j  stone    or  V  Pch. 
(  masonry,  ) 

SCALE— ascending,  1728,  27,  40,  50,  16,  8,  128,  24f ;  descend- 
ing, 24f,  128,  8,  16,  50,  40,  27,  1728. 

NOTES.    1.  A  cubic  yard  of  earth  is  called  a  load. 

2.    Railroad  and  transportation  companies  estimate  light  freight  by  the  space  it 
occupies  in  cubic  feet,  and  heavy  freight  by  weight. 


cubic  inches  (cu.  in.) 
cubic  feet 

cubic  feet  of  round  timber,  or ) 
"        "    hewn  J 

cubic  feet 
cord  feet,  or  ) 
cubic  feet      f 

cubic  feet 


MEASURES   OF  CAPACITY.  189 

3.  A  pile  of  wood  8  feet  long,  4  feet  wide,  and  4  feet  high,  contains  1  cord  ;  and 
a  cord  foot  is  one  foot  in  length  of  such  a  pile. 
4   A  perch  of  stone  or  of  masonry  is  16 >£  feet  long,  1>£  feet  wide,  and  1  foot  high. 

MEASURES  OF  CAPACITY. 

148.  Capacity  signifies  extent  of  room  or  space. 

All  measures  of  capacity  are  cubic  measures,  solidity  and 
capacity  being  referred  to  different  units,  as  will  be  seen  by 
comparing  the  tables. 

Measures  of  capacity  may  be  properly  subdivided  into 
two  classes,  Measures  of  Liquids,  and  Measures  of  Dry 
Substances. 

I.    LIQUID   MEASURE. 

149.  Liquid  Measure,  also  called  Wine  Measure,  is 
used  in  measuring  liquids;  as  liquors,  molasses,  water,  &c. 


TABLE. 

4    gills  (gi.)  make  1  pint, pt, 

2    pints  "       1  quart, qt. 

4    quarts  "       1  gallon, gal. 

81i  gallons  "      1  barrel, bbL 

2    barrels,  or  63  gal.  '«      1  hogshead, . .  .hhd. 

UNIT  EQUIVALENTS. 

Pt.  gl. 

qt  1—4 

gaL  1    —        2   —  8 

bbl         1—4—8—82 
hhd.       1  =  3H  —  126  —  252  —  1008 
1  _  2  =*  63     —  252  —  504  —  2G16 
SCALE — ascending,  4,  2,  4,  31$,  2;   descending,  2,  81$,  4,  2, 4* 


140  COMPOUND  NUMBERS. 

1«5O.  The  following  denominations  are  also  in  use: 

86  gallons  make!  barrel        of  beer. 

54      "        or  H  barrels          "      1  hogshead   "    •• 
42      "  "      1  tierce. 

2  hogsheads,  or  120  gallons,  <{      1  pipe  or  butt 

2  pipes,  or  4  hogsheads,          *'      1  tun. 

NOTES.    1.  The  denominations,  barrel  and  hogshead,  are  used  in  estimating  tne 
capacity  of  cisterns,  reservoirs,  vats,  &c. 

2.  The  tierce,  hogshead,  pipe,  butt  and  tun,  are  the  names  of  casks,  and  do  not 
express  any  axed  or  definite  measures.    They  are  usually  gauged,  and  have  their 
capacities  in  gallons  marked  on  them. 

3.  Ale  or  beer  measure,  formerly  used  hi  measuring  beer,  ale  and  milk,  is  almost 
entirely  discarded. 


II.     DRY  MEASURE. 

151.     Dry  Measure  is  used  in  measuring  articles  not 
liquid ;  as  grain,  fruit,  salt,  roots,  ashes,  &c. 

TABLE. 

2  pints  (pt.)  make  1  quart, . . .  .^" qt. 

8  quarts          "        1  peck, pk. 

4  pecks  "       1  bushel,. bu.  or  bush. 

UNIT  EQUIVALENTS. 

qt.  pt. 

pk.  1    -      2 

bu.       1  _    8  -  16 
1—4  —  82  —  64 
SCALE — ascending,  2,  8,  4;  descending,  4,  8,  2. 

NOTES.  1.  In  England,  8  bu.  of  70  Ibs  each  are  called  a  quarter,  used  in  measuring 
grain.  The  weight  of  the  English  quarter  is  ^  of  a  lon£  ton. 

2.  The  wine  and  dry  measures  of  the  same  denomination  are  of  different  capacl 
ties.  The  exact  and  the  relative  size  of  each  may  be  readily  Been  by  the  following 


TIME.  141 


COMPARATIVE  TABLE  OF  MEASURES  OF  CAPACITY. 

Cu.in.in        Cu.  in.  in        Cu.in.in        Cu.in.in 
one  gallon,     one  quart.       one  pint.          one  gill. 

Wine  measure,          231  57f          28£          7/2 

Drymeasnre;(*pk.,)268J        67|          33|          8| 

3.  The  b«er  gallon  of  282  inches  is  retained  in  nse  only  by  custom.    A  bushel 
commonly  estimated  at  2150.4  cubic  inches. 


MEASURE  OF  TIME. 
153.    Time  is  the  measure  of  duration. 


TABLE. 

60  seconds  (sec.)  make  1  minute, ^  .min. 

60  minutes  "  1  hour, ..h. 

24  hours  "  1  day, ...... da. 

7  days  «'  1  week, wk. 

365  days  "  1  common  year, ..... .yr. 

866  days  "  1  leap  year, yr. 

12  calender  months  '*  1  year, .yr. 

100  years  "  1  century, «-^0. 

CJN'IT  EQUIVALENTS. 

min.        sec. 

h.       1  -      60 

dn.     1  —    60  —    8600 

wk.      1  —  24  —   1440  —   88400 

1  —    7    168  —  10080  —   604800 

yr.   mo.      f  365  —  8760  —  525600  —  81536000 

1—12      (  366  —  8784  —  527040  —  31622400 

SCALE— ascending,  60,  60,  24,  7;  descending,  7,  24,  60,60. 


H2 


COMPOUND   NUMBERS. 


The  calendar  year  is  divided  as  follows  : — 


No  .  of  month  .      Season  . 

Names  of  months 

1 

2 

Winter, 

j  Jamiary, 
(  February, 

8 
4 
5 

Spring, 

(  March, 
1  April, 
(May, 

6 
7 
8 

Summer, 

(June, 
]  July, 
(  August, 

9 
10 
11 

Autumn, 

(  September, 
<  October, 
(  November, 

12 

Winter, 

December, 

Abbreviations.      No.  of  days. 


Jan. 
Feb. 

Mar. 
Apr. 

Jun. 

Aug. 

Sept. 

Oct. 

Nov. 

Dec, 


31 

28  or  2* 

81 
30 
31 

30 
81 
81 

30 
31 
80 

81 


865or3i/6 


NOTES.  1.  The  exact  length  of  a  solar  year  is  365  da.  5  h.  48  min.  46  sec.  ;  but 
fbr  convenience  it  is  reckoned  11  min.  14  sec.  more  than  this,  or  365  da.  6  h.  — 
265>£  da.  This  %  day,  in  four  years  makes  one  day,  which,  every  fourth,  bissex- 
tile, or  leap  year,  is  added  to  the  shortest  month,  giving  it  29  days.  The  leap  year* 
are  exactly  divisible  by  4,  as  1856, 1860, 1864. 

The  number  of  days  in  each  calendar  month  may  be  easily  remembered  by 
committing  the  following  lines  :— 

"  Thirty  days  hath  September, 
April,  June,  and  November  ; 
All  the  rest  have  thirty -one, 
Save  February,  which  alone 
Hath  twenty-eight ;  and  one  day  more 
We  add  to  it  one  year  in  four." 

2.  In  most  business  transactions  30  days  are  called  1  month. 

3.  The  centuries  are  numbered  from  the  commencement  of  the  Christian  era  , 
the  months  from  th«  commencement  of  the  year  ;  the  days  from  the  commence- 
ment of  the  month,  and  the  hours  from  the  commencement  of  the  day,  (12  o'clock, 
midnight.)    Thus,  May  23d,  I860,  9  o'clock  A.  M.,  is  the  9th  hour  of  the  23d  day 
of  the  5th  month  of  the  60th  year  of  the  19th  century. 


CIRCULAR   MEASURE.  143 

CIRCULAR   MEASURE. 

155.  Circular  Measure,  or  Circular  Motion,  is  used 
principally  in  surveying,  navigation,  astronomy,  and  geogra- 
phy, for  reckoning  latitude  and  longitude,  determining  loca- 
tions of  places  and  vessels,  and  computing  difference  of 
time. 

Each  circle,  great  or  small,  is  divisible  into  the  same 
number  of  equal  parts,  as  quarters,  called  quadrants, 
twelfths,  called  signs,  360ths,  called  degrees,  &c.  Conse- 
quently the  parts  of  unequal  circles,  although  having  the 
same  names,  are  of  unequal  lengths. 

TABLE. 

60  seconds  (")  make  1  minute,. . . . '. 

60  minutes  u      1  degree, . . . .  °. 

80  degrees  "      1  sign, S. 

12  signs,  or  360°      "       1  circle, 0. 

UNIT  EQUIVALENTS. 

t        n 
1  —     60 

a  1  —  60  —  3600 
0.  1  —  30  —  1800  —  108000 
1  —  12  —  860  —  21600  —  1296000 

SCALE — ascending,  60,  60,  30,  12  ;  descending,  12,  30,  60,  60. 

NOTES.  1.  Minutes,  of  the  earth's  circumference  are  called  geographic  or  nauti- 
cal miles. 

2.  The  denomination,  signs,  is  confined  exclusively  to  Astronomy. 

8.  A  degree  has  no  fixed  linear  extent.  When  applied  to  any  circle,  it  is  alwayi 
•j-g--g-  part  of  the  circumference.  But,  strictly  speaking,  it  is  not  any  part  of  a 
circle. 

4.    90*  make  a  quadrant  or  right-angto. 

5     60"  make  a  soxtaat  or      of  a  circl*. 


144:  COMPOUND   NUMBERS. 

MISCELLANEOUS  TABLES. 

156.    COUNTING. 

12  units  or  things  make  1  dozen. 
12  dozen  "     1  gross. 

12  gross  "     1  great  gross. 

20  units  t4     1  score. 

157.  PAPER. 

24  sheets  .....  malse  ....  1  quire. 
20  quires  1  ream. 

2  reams  1  bundle. 

5  bundles         "  1  bale. 


BOOKS. 

The  terms  folio,  quarto,  octavo,  duodecimo,  &c.,  indicate 
the  number  of  leaves  into  which  a  sheet  of  paper  is  folded. 

A  sheet  folded  in    2  leaves  is  called  a  folio. 

A  sheet  folded  in    4  leaves  "  a  quarto,  or  4to. 

A  sheet  folded  in    8  leaves  **  an  octavo,  or  8vo. 

A  sheet  folded  in  12  leaves  "  a  12mo. 

A  sheet  folded  in  16  leaves  "  a  16mo. 

A  sheet  folded  in  18  leaves  ,.  M  an  18mo. 

A  sheet  folded  in  24  leaves  "  a  24mo. 

A  sheet  folded  in  32  leaves  *  a  32mo. 


COPYING. 

75  words  make  1  folio  or  sheet  of  common  law. 
90      "        "      1     "     "      "     "  chancery. 
1  60.  An  Aliquot  Part  of  a  number  is  such  a  part  as 
will  exactly  divide  that  number;  thus,  3,  5,  7£  are  aliquot 
parts  of  15. 

NOTE.     An  aJ*<ptat  part  may  bo  a  whole  or  mixed  number,  while  A  factor  must  b« 
»  whole  number. 


ALIQUOT    PARTS. 


145 


ALIQUOT  PARTS  OF  ONE  DOLLAR. 


161. 

50  cents  =  J  of  1  dollar. 
33i  cents  =  $  of  1  dollar. 
25  cents  =  i  of  1  dollar. 
20  cents  =  1  of  1  dollar. 
16  §  cents  =  i  of  1  dollar. 


12*  cents  =  £  of  1  dollar. 
10    cents  =03  of  1  dollar. 

8i  cents  =  T'5  of  1  dollar. 

6i  cents  =T^  of  1  dollar. 

5    cents  ='   of  1  dollar. 


1  63.   PARTS  OF  81  IN  NEW  YORK  CURRENCY. 


4  shillings  = 

2  shillings  8d.        = 
2  shillings  = 


1  shil.  4  pence        = 
1  shilling  = 

6  pence  = 


1  63.  PARTS  OF  $1  IN  NEW  ENGLAND  CURRENCY. 


3  shillings 
2  shillings 
1  shillings  6d. 

164. 

10  hund.  Ibs.    = 


1  shilling 
9  pence 
6  pence 


J  ton. 

5  hund.  Ibs.    =  J  ton. 
4  hund.  Ibs.    =  i  ton. 


ALIQUOT  PARTS  OF  A  TON. 

2  hund.  2  qrs. 
2  hund.  Ibs. 
1  hund.  Ibs. 


=  Si 


=     ton. 


1  63.     ALIQUOT  PARTS  OF  A  POUND  AVOIRDUPOIS. 

8  ounces       =  \  pound.     I     2  ounces        —  |-  pound. 
4  ounces       =  {  pound.          1  ounce         =T1g  pound. 


166. 

ALIQUOT  PARTS  OF  TIME. 

Parts  of  1  year. 

Parts  of  1  month. 

6 

months 

— 

2  year. 

15 

days 

=  £     month. 

4 

months 

— 

i  year. 

10 

days 

=  J     month. 

3 

months 

— 

i  year. 

6 

days 

=  i     month. 

2 

months 

= 

i  year. 

5 

days 

=  i     month. 

li 

mouths 

— 

I  year. 

3 

days 

=r  y1^  month. 

1J 

months 

^^ 

J  year. 

2 

days 

zn  y'-j  month. 

1 

month 

=  ^2  year. 

1 

day 

=  -^Q  month 

146  COMPOUND   NUMBERS. 

REDUCTION. 

1 67.  Reduction  is  the  process  of  changing  a  number 
from  one  denomination  to  another  without  altering  its  value. 

168.  Reduction  Descending  is  changing  a  number  of 
one  denomination  to  another  denomination  of  less  unit  value, 
and  is  performed  by  multiplication  ;  thus :  $1  =  10  dimes 
=  100  cents  =  1000  mills;  1  yard  =  3  feet  —  36  inches. 

1.  Reduce  6  gal.  2  qt.  1  pt.  to  pints. 

OPERATION.  ANALYSIS.     Since  in  1  gal. 

6  gal.  2  qt.  1  pt.          there  are  4   qt.  in  6  gal.  there 
4  4  qt.X6=-24  qt     and  the  2 

qt    in  the  given  number,  ad-.. 

26  <!*•  ded,  makes  26  qt  in  6  gal.  2  ; 

qt       Since  in  1  qt.  there  are  ' 

Ans.  53  pt  2  ^  in  26  &  there  2  Pt  X 

26=52  pt    and  the  1  pt  in 

the  given  number  added,  make  53  pints  in  the  given  com- 
pound number.  As  either  factor  may  be  used  as  a  multipli- 
cand, (  61  )i  we  may  consider  the  numbers  in  the  descend- 
ing scale  as  multipliers.  Hence  the  following 

RULE.  I.  Multiply  the  highest  denomination  of  the  giver* 
compound  number  ~by  that  number  of  the  scale  which  will 
reduce  it  to  the  next  lower  denomination,  and  add  tj  the 
product  the  given  number,  if  any ,  of  that  lower  denomination . 

II.  Proceed  in  the  same  manner  with  the  result  obtained 
in  each  lower  denomination,  until  the  reduction  is  brought  to 
the  denomination  required. 

EXAMPLES    FOR   PRACTICE. 

2.  In  8  Ib.  10  oz.  how  many  ounces  ?        Ans.  138  oz. 

3.  In  £12  6s.  9d.  how  many  pence  1          Ans.  2961d. 
' ,  In  4  yd.  1  ft.  10  in.  how  many  inches  1 

Tn  3  mi.  5  fur.  26  rd.  how  many  rods  1 


REDUCTION.  147 

6.  In  18s.  8d.  3  far.  bow  many  farthings  1 

Ans.  899  far. 

7.  Reduce  3  Ib.  9  oz.  12  pwt.  to  pennyweights. 

8.  In  hhd.  15  gal.  2  qt.  how  many  pints  ? 

9.  Reduce  4  da.  5  hr.  to  minutes.       Ans.  6060  min. 

10.  Reduce  10  bu.  1  pk.  6  qt.  to  pints.     Ans.  1308  pt. 

11.  Reduce  14  A.  3  R.  20  sq.  rd.  to  square  rods. 

12.  Reduce  4  cd.  3  cd.  ft.  9  cu.  ft.  to  cubic  inches. 

13.  Reduce  4  yr.  7  mo.  to  hours.          Ans.  39600  hr. 

14.  Change  2  T.  11  cwt.  to  pounds.        Ans.  5100  Ib. 

15.  Change  U  Ib.  9  oz.  10  pwt.  to  grains. 

16.  Change  5  lb.-6  §  43  23  10  gr.  to  grains. 

17.  Change  3  mi.  6  fur.  to  feet.  Ans.  19800  ft. 

18.  In  40  chains  how  many  links  1  Ans.  4000  1. 
X19.  In  28  sq.  rd.  12  sq.  yd.  4  sq.  ft.  how  many  square 
inches'?  .                                             Ans.  1113840  sq.  in. 

20.  In  16  A.  4  sq.  ch.  8  P.  80  sq.  1.  how  many  square 
links?  Ans.  1645080  sq.  1. 

21.  In  12  tons  of  round  timber  how  many  cubic  inches  7 

22.  In  8  bbl.  26  gal.  how  many  pints  ?   Ans.  2224  pt. 

23.  Reduce  4  pipes  to  quarts.  Ans.  2016  qt. 
\24.  Reduce  23  bu.  3  pk.    to  pints.  Ans.  1520  pt. 

.  Reduce  8  S.  18°  40'  to  minutes.         Ans.  15520'. 

26.  Reduce  15°  to  seconds.  Ans.  54000". 

27.  Reduce  2  months  to  minutes.       Ans.  86400  min. 

28.  Change  2  reams    10  quires  to  sheets. 

29.  In  40  score  how  many  single  things'?     Ans.  800. 

30.  In  14  great  gross  how  many  dozens  ? 
In  30°  20'  24"  how  many  seconds  ? 

32.  In  the  8  Autumn  months  how  many  hours  1 

33.,  In  the  three  Summer  months  how  many  minutes  ? 

34.  In  75  cords  how  many  cubic  feet  ? 


148  COMPOUND    NUMBERS. 

169.  Reduction  Ascending  is  changing  a  number  of 
one  denomination  to  another  of  greater  unit  lalue,  and  is 
performed  by  Division;  thus,  1000  mills  =100  cents  =$1. 

1.  Reduce  53  pints  to  gallons. 

OPERATION.  ANALYSIS.     Dividing  the  given 

2)53  number  of  pints  by  2,  because  - 

there  are  ^  as  many  quarts  as 

4)26  qt.+l  pt.  pintS)  we  Oktain  26  qt.  plus  a  re- 

~"7      i  _i_o     f  mainder^of  1  pt.     We  next  divide 

26  qt,  .by  4,  because  there  are  | 

Ans.  6  gal.  2  qt.  1  pt  as  many  £allon£as  quarts,  and  wo 
obtain  6  gal.  and  a  remainder  of  2  qt.  Th^  lastjquotient,  with 
the  several  remainders  annexed,  forms  the  answer. 

2.  Reduce  4902  inches  to  rods. 

OPERATION.  ANALYSIS     We  divide  suc- 

12)4902  cessively  by  the  numbers  in 

the  ascending  scasTe  in  the 

16|)408  ft.-j-6  in.  same  manner  as  in  the  pre- 

ceeding  example.    But  in  di- 

~  viding  the  408  ft.  by  16^, 

'  _  we    first  reduce  408   ft.   to 

24  rd.-f-27y4=12  ft.      halves  by  multiplying  by  2, 

and  we  have  816  halves  ;  and 
Am.  24  rd.  12  ft.  6  in.       rcducing  16,  to  m^  we 


have  33  hakes.  Then  dividing  816  by  33  we  obtain  24  rd. 
plus  a  remainder  of  24  halves—  to  12  ft.  which,  with  the  proceed- 
ing remainder  annexed  to  the  last  quotient,  gives  the  answer. 

RULE.  I.  Divide  the  given  number  Ly  that  number  of 
Hie  scale  which  wilt  reduce  it  to  the  next  higher  denomina-. 
lion. 

II.  Divide  the  quotient  by  the  next  higher  number  in 
scale  j  and  so  proceed  to  //><•;  highest  denomination  required* 
The  last  qi/of?i')if}  with  the  several  remainders  annexed  in  a 
reversed  order,  will  be  the  ansiccr. 


REDUCTION.  149 

* 

EXAMPLES   FOR   PRACTICE. 

3.  How  many  pounds  in  3460  ounces  ? 

Ans.  216  Ib.  4  oz. 

4.  How  many  shillings  in  556  farthings  ? 

Ans.  11s.  7d. 

5.  "How  many  yards  in  1242  inches  1 

6.  How  many  gallons  in  2347  pints  ? 
7..  Reduce  23547  troy  grains  to  pounds. 

Ans.  4  Ib.  1  oz.  1  pwt.  3 
8.  Reduce  1597  quarts  to  bushels. 

Ans.  49  bu.  3  pk.  5  qt. 
9r  Reduce  107520  oz.  avoirdupois   to  pounds. 

10.  In  28635  sec.  how  many  hours  ? 

Ans.  7  hr.  57  min.  15  sec. 

11.  In  10000"/ow  many  degrees  ?        & 

'Ans.' 2°  46'  40". 
ftl2.  In  11521  gr.  apothecaries  weight  how  many  pounds  ? 

Ans.  21b  1  gr- 
°  13.  In  3561829  seconds  how  many  weeks? 

14.  Reduce  67893  cu.  ft.  to  cords. 
v  15.  In  1491  pounds  how  many  hundred  weight? 
5  16.  In  12244  pints  how  many  hogsheads  ? 

17.  In  25600  sq.  rd.  how  many  acres?      Ans.  160  A. 

18.  How  many  miles  in  51200  rd.  ?          Ans.  160  mi. 

19.  How  many  barrels  in  6048  gills?         Ans.  6  bbl. 

20.  In  316800  inches  how  many  miles  ?      Ans.  5  mi. 

21.  In  1728  how  many  gross  ?  Ans.  12  gross. 

22.  In  4060  how  many  score  ?  Ans.  203  score. 
A23.  Reduce  1435  feet  to  fathoms. 

24.  Reduce  10000  sheets  of  paper  to  reams. 

Ans.  20  reams  16  quires  16  sheets. 

25.  Reduce  27878400  sq.  ft.  to  square  miles. 


150  COMPOUND  NUMBERS. 

•au 
PROMISCUOUS    EXAMPLES    IN   REDUCTION. 

1.  Reduce  4  dollars  67  cents  to  cents.     Ans.  467  cents. 

2.  Reduce  3724  mills  to  dollars.  Ans.  $3.724. 

3.  Reduce  9690  cents  to  dollars.  Ans.  $96.90. 

4.  Reduce  8  dollars  to  mills.  Ans.  8000  mills. 

5.  In  91751  farthings  how'iSkft^painKfe^          ** 

Ans.  £95  Us.  5d^3  far. 
>6.  In  3  Ib.  4  oz.  7  pwt.  how  many  grains  1 
«^7.  In  3  tons  of  cheese  how  many  pounds  ? 

8.  How   much  will   4    cheese  cost,   each   weighing  36 
pounds,  at  9  cents  a  pound  1  Ans.  $12.96. 

9.  How  much  would  2  Ib.  8  oz.  12  pwt.  of  gold  dust  be 
worth,  at  72  cents  a  pwt.  ?  Ans.  $409.44. 

10.  Bought  1  T.  15  cwt.,36  Ib.  o£  sugar  at  7  cents  a 
pound;  howNjtoh  di^jtcost?  >'     .  ,N^.4ws.  $247.52. 

11.  Paid  $25,$rf<Sr  otnr  ho|^acf  of  molasses,  and  sold 
it  all  at  50  cents  a  gallon  ;  how  much  was  the  whole  gain  ^ 

VL2.  How  many  pounds  in  t>  barrels  of  flour  ? 

13.  How  many  bushels  of  oats  in  a  load  weighing  1280 
pounds]  Ans.  40  bu. 

14.  How  many  bushels  of  wheat  in  a  load  weighing  2175 
pounds  ?  Ans.  36  bu.  15  Ib. 

15.  A  grocer  bought  3  barrels  of  flour  at  $6  a  barrel, 
and  sold  it  out  at  4  cents  a  pound  •  how  much  did  he  gain 
on  the  whole  ?  Ans.  $5.52. 

\16.  In  a  board  12  feet  long  and  2  feet  wide,  how  many 
square  feet?  Ans.  24  sq.  ft. 

"    17.  In  a  block  of  marble  6  feet  long  and  3  feet  square, 
how  my  cubic  feet?     ^  Ans.  54  cu.  fect.-» 

18.  In  a  pile  of  woocT^  feet  long  6  feet  high  and  3  feet 
wide,  how  many  cubic  feet  ?  how  many  cords  ? 

'\  C\l1  \  Ans.  468  cu.  ft.  ;  or  3  Cd.  84  cu.  ft. 


c 


REDUCTION.  151 

19.  In  259200  cubic  inches  of  hewn  timber  how  many 
tons  ?  Ans.  3  T. 

^^0.  How  many  square  rods  in  a  field  90  rods  long  and  75 
rods  wide  ?     How  many  acres  ?       Ans.  42  A.  30  sq.  rd. 

21.  A  pond  ot  water  measures  3  fathoms  2  feet  9  inches 
in  depth ;  how  many  inches  deep  is  it  ?         Ans.  249  in. 

22.  What  will  3  miles  of  telegraph  cable  cost  at  12  cents 
afoot?  Ans.  $1900.80. 

23.  What  is  the  age  of  a  man  3  score  and  5  years  old  1 

Ans.  65  years. 

24.  How  much  will  I  receive  for  a  load  of  wheat  weigh- 
ing 2760  pounds  at  $1.50  per  bushel  1  Ans.  $69. 

25.  How  many  cubic  feet  in  a  stick  of  timber  32   feet 
long  2  feet  wide  and  1  foot  thick  ?  Ans.  64  cu.  ft. 
\26i>   How  many  square  feet  in  one  acre  ? 

^  1^7.  In  176  yards  how  many  rods  ?  Ans.  32  rd. 

28.  A  pile  of  wood  is^lCjJ^jJgjjg,  8  feet  high,  and  8 
feet'  w"k!ef  now  much  is  it  worth  at  $3.50  a  cord  ? 

Ans.  $28.    ' 

29.  What  would  be  the  value  of  a  city  lot  40  feet  wide 
and  120  feet  long,  at  2  cents  a  square  foot  ?      Ans.  $96. 

80.  A  grocer  bought  4  barrels  of  cider,  at  $2  a  barrel,  and 
after  converting  it  into  vinegar,  he  retailed  it  at  15  cents  a 
gallon ;  how  much  was  his  whole  gain.  Ans.  $10.90. 

31.  At  6  cents  a  pint  how  much  molasses  can  be  bought 
'for  $4.26?  Ans.  8  gal.  3  qt.  1  pt. 

32.  An*  innkeeper  bought  a  load  of  40  bushels  of  oats, 
at  36  cents  a  bushel,  and- retailed  them  at  25  cents  a  peck  j, 
how  much  did  he  make  on  the  load?  Ans.  $25.60. 

23.  What  will  be  the  cost  of  a  hogshead  of  wine  at  8 
cents  a  gill?  Ans.  $161.28. 

34.  In  120  gross  how  many  score  ?        Ans.  864  score. 


152  COMPOUND  NUMBERS. 

85.  If  a  man  walk  4  miles  an  hour,  and  10  hours  a  day,  • 
how  many  miles  can  he  walk  in  24  days?    Ans.  960  mi. 

26.  What  will  be  the  cost  of  2  bu.  1  pk.  G  qt.  of  tiinj£ 
thy  seed,  at  10  cents  a  quart?  •  Ans.  $7.80. 

87.  What  would  be  the  value  of  a  silver  goblet,  weigh-     -^ 
ing  8  oz.  14  pwt.,  at  $.15  a  pwt.  ?  Ans.  $26.10. 

88.  What  .will  16  reams  of  paper  cost  at  20  cents  a  quire!     ^ 

Ans.  $64. 

39.  If  1  bushel  of  wheat  make  45  pounds  of  flour,  how   ^— 
many  pounds  will  500  bushels  make  ?     How  many  barrels  ] 

Ans.  114  bbl.  156  pounds. 

40.  Bought  a  gold  chain,  weighing  2  oz.  18  pwt.  at  $.90 
a  pwt.;  how  much  did  it  cost?  Ans.  $52.20. 

41.  How  many  minutes  more.iare  there  in  the  Summer 

than  in  the  Autumn  months  ?  Ans.  1440  min.X 

t^   j 

\^  42.     How  much  will  it  cost  to  dig  a  -cellar  24  ft;  long; 
18  ft.  wide  and  6  feeUkeD^ULcent  a  cufoc  Foot ? 

^^^^^^^^^^^^^^"*^**  • 


"-  43.  How  many  boxes,  each  containing  12  pounds,  can  bt 
filled  from  a  hogshead  of  sugar  containing  9  cwt.? 

Ans.  75  boxes. 

*^  44.  What  will  be  the  cost  of  5  bales  of  cloth,  each  bak 
containing  15  pieces,  and  each  piece  measuring  26  yards, 
at  $1.75  a  yard? 

s^.45.  If  a  cannon  ball  goes  at  the  rate  of  10  miles  a  min-j 
ute,  how  many  miles  would  it  go,  at  the  same  rate,  in  2 
hours?  Ans.  12QO  miles. 

*^  46.  At  11  cents  a  pound  what  will  be  the  cokt  of  3  cwt. 
"2  qr.  21  Ib.  of  coffee?  ^te.\$40.81. 

47.  If  a  man  earn  $30  a  month,  how  much  will  he  earn 
in  5  years?  ^^Ans.  $1800. 

r^ 
W 


ADDITICtff.  '  153  - 

« 

ADDITION.  - 

1 7O.  Compound  numbers  are  added,  subtracted,  multi- 
plied, and  divided  by  the  same  general  methods  as  are  em- 
ployed in  simple  numbers.  The  only  modification  of  the 
operations  and  rules  is  that  required  for  borrowing,  carry- 
ing, and  reducing  by  a  vaiying,  instead  of  a  uniform  scale. 

1.  What  is  the  sum  of  36  bu.  2  pk.  6  qt.  1  pt.,  25  bu. 
1  pk.  4  qt.,  18  bu.  3  pk.  7  qt.  1  pt.,  9  bu.  Opk.^^t.  1'pU 

OPERATION.  ^  AyjA'siM   AiTHnging 


*«•         ?*•        qt.         pt.         the  rtumbere  in  columns, 
or;          -i          4          A          plaei.ig  units  of  the  same 


o          7          i  draTO  under  each 

021  ,"»we   first  iukl    the^ 

units  in   the   right  hand 


Ans.  90  0.  4  1  v,.mn,  otiowest  denom- 

ination, and  find  the 

amount  to  be  3  pints,  which  is  equal  to  1  qt  1  pt.  We  write 
the  1  pt.  under  the  column  of  pints,  and  add  the  i  qt.to  the  col- 
umn of  quarts.  We  find  the  amount  of  the  second  column  to 
be  20  qt.  which  is  equal  to  2  pk.  4  qt  Writing  the  4  qt  under 
the  column  of  quarts,  we  add  the  2  pk.  to  the  column  of  pecks. 
Adding  the  column  of  pecks  in  the  same  manner,  we  find  the 
amount  to  be  8  pk.  equal  to  2  bu.  Writing  0  pk.  under  the  col- 
umn of  pecks,  we  add  the  2  bu.  to  the  column  of  bushels.  Add- 
ing the  last  column,  we  fird  the  amount  to  be  90  bu.  which  we 
write  under  the  left  hand  denomination,  as  in  simple  numbers. 
Hence  the  following 

RULE  1.  Write  the  numbers  so  tliat  tlwse  of  the  same 
unit  value  will  stand  in  tJie  same  column. 

II.  Beginning  at  tJie  riylit  hand,  add  each  denomination 
as  in  simple  numbers,  carrying  to  each  succeeding  denomi- 
nation one  for  as  many  units  as  it  takes  of  the  denomination 
added  j  to  make  one  of  the  next  higher  denomination. 


154 

COMPOUND  NUMBERS. 
EXAMPLES  FOR  PRACTICE. 

(2.) 

(3.) 

£. 

S. 

d. 

far. 

ft) 

z 

3  . 

'f)  er 

47 

10 

9 

1 

10 

10 

4 

1  12 

25 

6 

4 

3 

9 

5 

2  10 

36 

18 

0 

2 

14 

4 

0 

0  16 

12 

00 

10 

0 

6 

0 

P-r 

7 

1  00 

8 

7 

3 

1 

6 

3 

2  15 

Ans.lSQ 

3 

3 

3 

32 

7 

5 

2  13 

(4)                 (5) 

hhd. 

gal. 

qt. 

pt. 

T.  cwt.  Ib. 

oz. 

dr. 

24 

21 

3 

1 

3  12 

15 

10 

11  • 

102 

42 

2 

0 

16 

20 

7 

9 

38 

9 

0 

1 

5   9 

6 

0 

12 

42 

50 

1 

0 

18 

17 

14 

00 

207 

60 

3 

0 

10  15 

59 

1 

00 

(6) 

(7) 

da. 

h. 

min. 

sec. 

Ib.  oz. 

pwt. 

gr. 

27 

14 

40 

36 

16 

11 

18 

.21 

106 

%20 

14 

25 

26 

9 

15 

10 

16 

12 

50 

45 

11 

10 

00 

8 

52 

16 

39 

18 

4 

6 

12 

00 

(8) 

(9) 

mi. 

fur. 

rd. 

yd.  ft. 

in. 

P. 

sq.yd.  sq.ft. 

2 

5 

25 

4   1 

10 

12 

20 

5 

1 

3 

30 

1   2 

7 

9 

15 

6 

4 

0 

16 

5   0 

4 

15 

10 

7 

10 

6 

8 

2   2 

11 

20 

26 

3 

ADDITION.  155 

—  10.  What  is  the  sum  of  2S.  12°,  40',  25";  5S.  9°,  27', 
88"; 16°   10'  50";  IS,  16°? 

11.  What  is  the  sum  of  44A.  2E.  24P.,  10A.  OE.  20P., 
25A.  IE.   6^.  36P.?  Ans.  86A.  IE. 

12.  What  is  the  sum  of  25  rd.  12  ft.  5  in.,  28  rd.  9  ft 

10  in.,  18  rd.  10  ft.,  12  rd.  14  ft.  9  in.? 

Ans.  2  fur.  5  rd.  14  ft. 

13.  What  is  the  sum  of  5  Cd.  6  cd.  ft.  9  cu.  ft.,  4  Cd.  3 
cd  ft.  12  cu.  ft.,  10  Cd.  14  cu.  ft.,  2  Cd.  7  cd.  ft.? 

Ans.  23  Cd.  2  cd.  ft.  3  cu.  ft. 

^14.  What  is  the  sum  of  40  yd.  2  ft.  10  in.,  37  yd.  1  ft.  9 
in.,  28  yd.  11  in.,  10  yd.  2  ft.,  15  yd.  ? 

Ans.  132  yd.  1  ft.  6  in. 

•^15.  What  is  the  sum  of  13  Cd.  60  cu.  ft.  164  cu.  in.,  25 
Cd.75  cu.  ft.,  18  Cd^25  cu.  ft.  540  cu.  in.,  8  Cd.  1030  cu. 
in.?  Ans.  65  Cd.  33  cu.  ft.  6  cu.  in. 

\^16.  A  grocer  bought  4  hhd.  of  sugar ;  the  first  weighed 

11  cwt.  2  qr.  21  lb.;  the  second  10  cwt.  1  qr.  16  lb.;  the 
third  10  cwt.  22  lb.;  and  the  fourth  9  cwt.  3  qr.     How 
much  did  the  whole  weigh  ?  Ans.  2T.  2  cwt.  9  lb. 

«^17.  A  man  has  a  farm  divided  into  three  fields;  the  first 
-^contains  26  A.  2  E.  30  P. ;  the  second,  48  A.  27  P. ;  an 
the  third,  35  A.  2  E.     How  many  acres  in  the  farm  ? 

Ans.  110  A.  1  E.  17  P. 

18.  If  a  printer  one  day  use  2  bundles  1  ream  10  quires 
of  paper,  the  next  day  3   bundles  1  ream  12  quires,  20 
sheets,  and  the  next,  4  bundles  9  quires,    how  much  does 
he  use  in  the  three  days  ? 

Ans.  10  bundles  1  ream  11  quires  20  sheets. 

19.  A  tailor  used,  in  one  year,  3  gross  6  doz.  10  buttons, 
another  year,   2   gross  9  doz.  9  buttons,  and  another  year, 
4  gross  7  doz. ;  how  many  did  he  use  in  the  three  years'? 


156  COMPOUND  NUMBERS. 

SUBTRACTION. 
171.  From  24  Ib.  6  oz.  5  pwt.  12  gr.  take  14  Ib.  9  oz. 

10  pwt.  7  gr. 

OPERATION.  ANALYSIS.    Writing  the 

oz.       pwt.        gr.         subtrahend     under     the 
6         5       12  .         ,     ,    .          ..     c 

Q       -.  p.          -          minuend,  placing  units  or 

the    same    denomination 

Ans.  98       15          5         under  each  other,  we  sub- 
tract 7  gr.  from  12   gr. 

and  write  the  remainder,  5  gr.,  underneath.  Since  we  cannot 
subtract  10  pwt.  from  2  pwt,  we  add  1  oz.  or  20  pwt.  to  the^. 
5  pwt.  and  subtract  10  pwt  from  the  sum, 25  pwt,  and  write 
the  remainder,  15  pwt,  underneath.  Having  added  20  pwt  or 
1  oz.  to  the  minuend,  we  now  add  1  oz.  to  the  9  oz.  in  the  sub-^ 
trahend,  making  10  oz ;  but  as  we  cannot  take  10  oz.  from  6  oz. 
we  add  1  Ib,  or  12  oz.  to  the  6  oz.  making  18  oz.  and  subtract- 
ing 10  oz.  from  18  oz.  we  write  the  remainder,  8  oz.  under  the 
denomination  r*  ounces.  Having  added  1  Ib.  to  the  minuend, 
we  now  add  1  ib.  to  the  14  Ib.  in  the  subtrahend,  and  subtract- 
ing 15  Ib.  from  24  Ib.  as  in  simple  numbers,  we  write  the  re- 
mainder, 9  Ib.  under  the  denomination  of  pounds.  Hence 

RULE.     I.    Write  the   subtrahend  under   the   minuend,*** 
so  that   units  of  the  same  denomination  shall  stand  under* 
each  other. 

II.  Beginning  at  the  right  hand,  subtract  each  denomi- 
nation separately,  as  in  simple  numbers. 

III.  If  the  number  of  any  denomination  in  the  subtra- 
hend exceed  that  of  the  same  denomination  in  the  minuend, 
add  to  the  number  in  the  minuend  as  many  units  as  make 
one  of  the  next  higher  denomination,  and  then  subtract  j  in 
this  case  add  1  to  the  next  higher  denomination  of  the  sub- 
trahend before  subtracting.     Proceed  in  the  same  manner 

mth  oac'h  denomination. 


SUBTRACTION.  157 

EXAMPLES  FOR  PRACTICE. 


(2)  (3) 

cwt.    qr.    Ib.  oz.  dr.  lihd.  gal.    qt.  pt. 

From  18       1     14  9  8  7     28      2  1 

Take     5      2     20  6  10  3     42      3  0 


12 

2 

19 

2 

14 

3  48 

3 

1 

(4) 

(5 

) 

fh 

K 

3  . 

•g 

°T 

bu.  pk. 

qt. 

pt. 

12 

7 

3 

1 

11 

104   2 

T. 

6 

Jr** 

0 

8 

5 

4 

2 

15 

56   3 

4 

1 

(6) 

(7) 

mi. 

fur. 

rd. 

yd. 

ft.  in. 

A. 

R. 

p. 

40 

5 

30 

3 

2  10 

400 

2 

25 

14 

6 

15 

4 

1  01 

325 

1 

30 

(8) 

(9 

) 

wk. 

da. 

hr.  i 

nin. 

sec. 

S.  ° 

r 

n 

10 

4 

16 

40 

22 

6  25 

45 

38 

4 

5 

12 

45 

50 

4  28 

40 

50 

(10) 

(11 

) 

T.  c 

!Wt. 

qr. 

Ib. 

oz. 

Cd.  cd.ft.< 

JU.ft. 

cu.i 

Q. 

14 

5 

2 

18 

9 

120   4 

6 

520 

10 

14 

3 

12 

14 

94   T 

12  1 

500 

0 

2) 

(13) 

(1J 

0 

yd. 

ft. 

in. 

Cd.  cu.ft 

eq.yd. 

sq.fl 

i  sq. 

in 

74 

2 

6 

325  80 

27  ' 

6 

91 

3 

9 

2 

9 

128  112 

14 

8 

12( 

) 

158  COMPOUND     NUMBERS. 

15.  From  125  mi.  6  fur.  take  90  mi.  4  fur.  25  rd. 

Ans.  35  mi.  1  fur.  15  rd. 

16.  A  man  bought  1  hhd.  of  molasses,  and  sold  42  gal. 
3  qt.  1  pt. ;  how  much  remained  1          Ans.  20  gal.  1  pt. 

17.  A  person  bought  9  T.   14  cwt.  3  qr,  of  coal,  and 
having  burned  4  T.  15  cwt.  sold  the  remainder ;  how  much 
did  he  sell  ?  Ans.  4  T.  19  cwt.  3  qr. 

18.  If   from  a  tub  of  butter  containing  1  cwt.  21   Ib 
there  be  sold  24  Ib.  8  oz.  how  much  remains  ? 

Ans.  96  Ib.  8  oz. 

19.  From  a  pile  of  wood  containing  42  Cd.  5  cd  ft.  there 
was  sold  16  Cd.  6  cd.  ft.  12  cu.  ft. ;  how  much  remained  ? 

Ans.  25  Cd.  6  cd.  ft.  4  cu.  ft. 

20.  If  from  a  field  containing  37  A.  3  R.  26  P.  there  be4 
taken  14  A.  2  R.  30  P.,  how  much  will  there  be  left  ? 

21.  A  farmer  having  raised  50  bu.  2  pk.  of  wheat,  kept 
for  his  own  use  25  bu.  3  pk.;  how  much  did  he  sell  ? 

Ans.  24  bu.  3  pk. 

22.  The  distance  from  New  York  to  Albany  is  150  miles; 
when  a  man  has  traveled  84  mi.  6  fur.  30  rd.  of  the  dis- 
tance, how  much  farther  has  he  to  travel  ? 

Ans.  65  mi.  1  fur.  10  rd. 

23.  What  is  the  difference  in  the  longitude  of  two  places 
one  71°  20'  26",  and  the  other  44°  35'  58"  West? 

Ans.  26°  44'  28". 

24.  If  from  a  'hogshead  of  molasses  10  gal.   2  qt.  be 
drawn  atone  time,  (T  gal.  3  qt.  at  another,  and  14   gal.  at 
another,  how  much  will  remain  ?  Ans.  28  gal.  3  qt. 

85.  From  a  section  of  land  containing  640  acres,  there 
was  sold  at  one  time  140  A.  2.  R.  36  P.,  at  another  time 
200  A.  1  R.,  and  at  another  time  75  A.  28  P. .  how  much 
remained  ?  Ans.  223  A.  3.  R.  16  P. 


MULTIPLICATION.  159 


MULTIPLICATION. 

172.  1.  A  farmer  lias  8  fields,  each  containing  4  A.  2 
R.  27  P.;  how  much  land  in  all  ? 

OPERATION.  ANALYSIS.     In  8  fields  are  8  times  as 

A-      R.      P.          much  land  as  in  1  field.     We  write  the 

'          multiplier  under  the  lowest  denomination 

of  the  multiplicand,  and  proceed  thus ;  8 

gy  -^  jg  times  27  P.  are  216  P.,  equal  to  5  R  16 
P.;  and  we  write  the  16  P,  under  the 
number  multiplied.  Then,  8  times  2  R.  are  16  R.,  and  5  R  ad- 
ded make  21  R.,  equal  to  5  A.  1  R ;  and  we  write  the  1  R  un- 
der the  number  multiplied.  Again,  8  times  4  A.  are  32  A.  and 
5  A.  added  make  37  A.,  which  we  write  under  the  same  de- 
nomination in  the  multiplicand,  and  the  work  is  done.  Hence 

RULE.  I.  Write  the  multiplier  under  the  lowest  denom- 
ination of  the  multiplicand. 

II.  Multiply  as  in  simple  numbers,  and  carry  as  in  ad- 
dition  of  compound  numbers. 

EXAMPLES   FOR   PRACTICE. 

(20  (3.) 

hhd.  gal.  qt.  pt.  bu.  pk.  qt.  pt. 

6   20   21  9261 

3  4 


Ans.  18   61   3   1  38    3   2  0 

(4.)  (5.) 

lb.   oz.  pwt.  gr.  T.   cwt.  Ib.  oz. 

12    8  14   16  10   15  20  8 

5  6 


63    7  13    8       64   14  23 


160  COMPOUND   NUMBERS. 

6.  Multiply  14  A.  2  R.  26  P.  by  8. 

AM.  117  A.  1  R.  8  P. 

7.  Multiply  6  yd.  2  ft.  9  in.  by  12.  Ans.  83  yd. 

8.  Multiply  7ft)-  8  §  .  5  3  .  13-  10  gr.  by  7. 

Ans.  54ft>.  0  |.  6  3. 1 3. 10  gr. 

9.  Multiply  24  bu.  1  pk.  6  qt.  by  10. 

10.  Multiply  9  cu.  yd.  15  cu.  ft.  520  cu.  in.  by  5. 

Am.  47  cu.  yd.  22  cu.  ft.  872  cu.  in. 

11.  Multiply  £84  12s.  6d.  2  far.  by  9. 

12.  If  a  pipe  discharge  4  hhd.  20  gal.  3  qt.  of  water  in 

1  hour,  how  much  will  it  discharge  in  5  hours,  at  the  same 
rate  ?  Am.  21  hhd.  40  gal.  3  qt. 

13.  If  a  load  of  coal  by  the  long  ton  weigh  1  T.  4  cwt. 

2  qr.  20  Ib.  what  will  be  the  weight  of  6  loads  ? 

Ans.  7  T.  8  cwt.  8  Ib. 

14.  If   1  acre  of  land  produce  26  bu.  3  pk.  4  qt.  of 
wheat,  how  much  will  11  acres  produce  ? 

15.  If  a  man  travel  30  mi.  4  fur.  20  rd.  in  1  day,  how 
far  will  he  travel  in  9  days,  at  the  same  rate  1 

,16.  What  is  the  weight  of  3   dozen  silver  spoons,  each 
dozen  weighing  2  Ib.  10  oz.  12  pwt.  14  gr.  ? 

Ans.  8  Ib.  7  oz.  17  pwt.  18  gr. 

17.  If  a  wood  chopper  can  cut  2  cd.  6  cd.  ft.  8  cu.  ft.  oi 
wood  in  a  day,  how  many  cords  can  he  cut  in  10  days  ? 

18.  In  20  barrels  of  potatoes,  each  containing  2  bu.  8 
pk.  6  qt.,  how  many  bushels  ?  Ans.  58  bu.  3  pk. 

19.  A  grocer  bought  14  barrels  of  sugar,  each  weighing 
5  cwt.  1  qr.  15  Ib.;  how  much  did  the  whole  weigh? 

20.  If  the  sun,  on  an  average,  change  his  longitude  59' 
9"  each  day,  how  much  will  be  the  change  in  25  days? 

21.  If  1  qt.  1  pt.  3  gi.  of  wine  fill  1   bottle,  how  much 
will  be  required  to  fill  3  dozen  bottles  of  the  same  capacity  ? 


MULTIPLICATION.  161 

22.  If  a  yard  of  cloth  cost  £2  10s.  9d.  how  much  will 
18  yards  cost  ?  Ans.  £45  13s.  6d. 

23.  If  a  person  average  8  hr.  20  min.  40  sec.  of  sleep 
daily,  how  much  will  he  sleep  in  30  days  ? 

Ans.  10  da.  10  hr.  20  min. 

24.  How  many  cords  of  wood  in  8  piles,  each  containin 
40  cd.  ft.  104  cu.  ft.  432  cu.  in.  1 

Ans.  46  Cd.  4  cd.  ft.  2  cu.  ft. 

25.  If  each  silver  table-spoon  weigh  1  oz.  12  pwt.  16  gr., 
what  is  the  weight  of  1  dozen  spoons  ? 

26.  If  the  moon's  average  daily  motion  is  33°  10'  35", 
how  much  of  her  orbit  does  she  traverse-  in  21  days  ? 

27.  How  much  land  in  12  lots,  each  containing  2  A.  120 
P.?  Ans.  33  A. 

28.  How  many  bushels  of  wheat  jn  48  sacks,  each  con- 
taining 165  pounds  ?  Ans.  132  bu. 

29.  If  a  locomotive  move  4  fur.  36  rd.  in  one  minute, 
how  far  will  it  move  in  one  hour  1        Ans.  36  mi.  6  fur. 

30.  If  a  family  consume  2  gal.  1  qt.  1  pt.  of  molasses  in 
1  week,  how  much  will  they  consume  in  1  year  ? 

Ans.  1  hhd.  60  gal.  2  qt. 

31.  If  it  take  a  man  5  hr.  42  min.  50  sec.  to  saw 
cord  of  wood,  how  long  will  it  take  him  to  saw  16  cords  ? 

Ans.  91  hr.  25  min.  20  sec. 

32.  How  many  bushels  of  apples  can  be  put  into  75  bar- 
rels, each  barrel  containing  3.  bu.  1  pk.  ? 

Ans.  243  bu.  3  pk. 

33.  If  a  man  can  build  3  rd.  12  ft.  10  in.  of  wall  in  1 
day,  how  much  can  he  build  in  10  days  ? 

I  Ans.  37  rd.  12^.  4  in. 

34.  If  a  man  can  mow  2  A.  96  P.  of  grass  in  a  day,  how 
much  can  27  men  mow,  at  the  same  rate? 
-    D  17; 
-.  <-  .  - 


162  COMPOUND  NUMBERS. 


DIVISION. 

173.  If  4  acres  of  land  produce  102  bu.  2  pk.  2  qt.  of 

wheat,  how  much  will  1  acre  produce  ? 

OPERATION.  ANALYSIS.     One  acre  will  pro- 

bn.    pk.  qt.   pta.        (iuce  J  as  much  as  4  acres.     Wri- 
4)102     3     2  ting  the  divigor  on  the  left  o{  tbe 

oc      2     ft     1          dividend,  we  divide  102  bu.  by  4, 
and  we  obtain  a  quotient  of  25  bu., 

and  a  remainder  of  2  bu.  We  write  the  25  bu.  under  tbe  de- 
nomination of  bushels,  and  reduce  the  2  bu.  'o  pecks,  making  8 
pk.,  and  the  3  pk.  of  the  dividend  added  makes  11  pk.  Divi- 
ding 11  pk.  by  4,  we  obtain  a  quotient  of  2  pk.  and  a  remain- 
der of  3  pk.  ;  writing  the  2  pk.  under  the  order  of  pecks,  we 
next  reduce  3  pk.  to  quarts,  adding  the  2  qt  of  the  dividend, 
making  26  qt,  which  divided  by  4  gives  a  quotient  of  6  qt.  and 
a  remainder  of  2  qt  Writing  the  6  qt.  under  the  order  of 
quarts,  and  reducing  the  remainder,  2  qt,  to  pints,  we  have  4 
pt,  which  divided  by  4  gives  a  quotient  of  1  pt,  which  w€ 
write  under  the  order  of  pints,  and  the  work  is  done. 

2.  A*  farmer  put  182  bu.  1        OPERATION. 
pk.  of  apples  into  46  barrels  546)  j^    P|v2  ^ 
how  many  bu.  did  he  put  in-     '   92 


40 
4 

When  the  divisor  is  largo  lfii(3 

we  divide  by  long  division,  as  -j^gg 

shown  in  the  operation.  From  _ 

these  examples  we  derive  the  23 
following  8 


184(4  qt. 

184 

-  Any.  2  bu.  3  pk.  4  qt. 


DIVISION.  163 

RULE.  I.  Divide  the  highest  denomination  as  in  simple 
numbers,  and  each  succeeding  denomination  in  the  same 
manner,  if  there  be  no  remainder. 

II.  If  there  be  a  remainder  after  dividing  any  denomina- 
tion, reduce  it  to  the  next  lower  denomination,  adding  in  the 
given  number  of  that  denomination,  if  any,  and  divide  as 
before. 

III.  The  several  partial  quotients  will  be   the  quotient 
required. 

EXAMPLES  FOR  PRACTICE. 
(3)  W 

A.     R.     P.  Ib.      oz.  pwt.   gr. 

2)95      2     30  3)52      4     16     18 


47      3     15  17      5    12      6 

(5) 
.     wk.  da.     h.  min.  sec.  bu. 

7)33       5     23    45     10  6)88 


4  5    20  32  10                     14  3  2 

(7)  (8) 

ft).     §•     3-  &•  gr.                   gal.  qt.  pt. 

5)28      9      4f  2  5               9)376  3  1 

5  9      0  2  17                    41  3  1 

(9)  (10) 

hhd.  gal.   qt.  pt.  A.  R.  P. 

12)9     28      2  0  9)129  2  25 


0  49   2   1  14   1  25 

(11)  (12) 

mi.  fur.  rd.  ft.  in.       Ib.   oz.  pwt.  gr 

7)217   5  19  12  6     11)185   1  19  13 


31   0  81   6   6   -     16   9  19  23 


164  COMPOUND   NUMBERS. 

13.  Divide  £185.  17s.  6d.  by  8. 

Ans.  £23.  4s.  8d.  1  far. 

14.  Divide  16  ft,.  13  oz.  10  dr.  by  6. 

Ans.  2  Ib.  12  oz.  15  dr. 

15.  Divide  358  A.  1  R.  17  P.  6  sq.  yd.  2  sq.  ft.  by  7. 

Ans.  51  A.  31  P.  8  sq.  ft. 

16.  Divide  192  bu.  3  pk.  1  qt.  1  pt.  by  9. 

Ans.  21  bu.  1  pk.  5  qt.  1  pt. 

17.  Divide  9  hhd.  28  gal.  2  qt.  by  12. 

•  .  ..     Ans.  49  gal.  2  qt,  1  pt. 

18.  Divide  328  yd.  1  ft.  3  in.  by  21. 

Ans.  15yd.  1  ft.  11  in. 

19.  Divide  36S>.  11  f.  4  3:  23.  7  gr.  by  11. 

Ans.  3R.  4  f  .  23.  13,  17  gr. 

20.  Divide  16  cwt.  3  qr.  18  Ib.,  long  ton  weight,  bj  32. 


21.  If  a  steamboat  run  174  mi.  26  rd.  in  14  hours,  how 
far  does  she  run  in  1  hour  ? 

22.  A  farm  containing  322  A.  2  R.  10  P.  is  to  by  divi- 
ded equally  among  13  persons  ;  how  much  will  each  .have  ? 

Ans.  24  A.  3  R.,  10  P. 

23.  A  cartman  drew  38  cd.  5  cd.  ft.  6  cu.  ft.  o'f  wood,  at 
80  loads  ;  how  much  did  he  average  per  load  ? 

Ans.  1  cd.  2  cd.  ft.  5  cu<*ft. 

24.  If  24  barrels  of  flour  cost  £98.  16s.,  how  much  will 
1  barrel  cost  1  Ans,  £4.  2s.  4d. 

25.  If  a  vessel  sail   163°  16'  12"  in  27  days,  how  far 
does  she  sail  on  an  average  per  day  ? 

Ans.  5°  40'  36". 

26.  If  3  dozen  spoons  weigh  9  Ib.  8  oz.  12  gr.,  how  much 
does  each  spoon  weigh  ?  Ans.  3  oz.  4  pwt.  11  gr. 


PKOMISCUOUS    EXAMPLES.  165 

PROMISCUOUS   EXAMPLES. 

1.  A  farmer  raised  200  bu.  2  pk.  of  barley,  175  bu.  3  pk. 
of  corn,  320    bu.  1  pk.  of  oats,  and  225  bu.  2   pk.  of  rye; 
what  was  the  whole  quantity  of  grain  raised  7 

2.  A  person   having  bought  325  A.  2   R.  of  land,  sold 
150  A.  1  R.  25  P.  of  it;  how  much  had  he  remaining? 

3.  What  is  the  whole   weight  of  72  hogsheads  of  sugar, 
each  weighing  12  cwt.  3  qr.  1  Ans.  45  T.  18  cwt. 

4.  If  a  railroad  car   run  148   miles  4  fur.  in  8  hours, 
what  is  the  average  rate  of  speed  per  hour  1 

5.  A  grocer   having  purchased   98    cwt.  2  qr.  of  sugar, 
sold  10  cwt.  1  qr.  20  Ib.  to  one  man,  and  18  cwt.  16  Ib.  to 
another;  how  much  remained  unsold-? 

6.  Bought  12  tea-spoons,  each   weighing  16  pwt.  20  gr., 
an'd  6  table-spoons,  each  weighing  1  oz.  12  pwt. ;  what  was 
their  total  weight  ?  Ans.  1  Ib.  7  oz.  14  pwt. 

7.  A  farmer  raised  24  T.  17  cwt.  of  hay;  he  sold  5  loads, 
each  weighing  1  T.  8  cwt.  21  Ib. ;  how  much  has  he  re- 
maining ?  Ans.  17  T.  15  cwt.  95  Ib. 

8.  A  jeweler  having  36  Ib.  10  oz.  14  pwt.  of  silver,  uses 
21  Ib.  6  oz.  of  it,  and  then  manufactures  the  remainder  into 
8  tea-pots ;  what  is  the  weight  of  each  ? 

Ans.  1  Ib.  11  oz.  1  pwt.  18  gr. 

9.  A  man  purchasing   2  A.  140  sq.  rd.  of  land,  reserves 
|  an  acre  for  his  own  use,  and  divides  the   remainder  in  4 
equal  lots ;  how  much  does  each  lot  contain  ? 

.4ns.  95  sq.  rd. 

10.  How  many  pounds  of  sugar  in  28  barrels,  each  con- 
taining 3  cwt.  1  qr.  17  Ib.  ?  Ans.  9576  pounds. 

11.  If  from  a  piece  of  land  containing   5  A.  3  R.,  2  A. 
72  P.  be  taken,  how  many  square  rods  will  remain  1 


166  COMPOUND   NUMBERS. 

12  Divide  a  tract  of  land  containing  1299500  square 
rods  into  25  farms  of  equal  area ;  how  many  acres  will 
there  be  in  each  ? 

Ans.  324  A.  3  R.  20  P. 

13.  A  merchant  buys  3  hogsheads  of  molasses  at  30  cents 
a  gallon,  and  sells  it  at  45  cents ;  how  much  does  he  gain 
on  the  whole  ? 

14.  What  is   the  cost  of  3  chests  of  tea,  each  weighing 
2  cwt.  2  qr.  18  lb.,  at  $.84  a  pound  ?  Ans.  225.12. 

15.  How   many  steps  of  30  inches  each   must  a   person 
take  in  walking  12  miles? 

16.  If  a  man  buy  10  bushels  of  chestnuts,  at  $3  a  bushel, 
and  sell   them  at   10  cents  a   pint,  how  much  is  his  whole 
gain?  Ans.  $34. 

17.  How  many  times  will  a  wheel  13  ft.  4  inches  in  cir- 
cumference turn  round  in  going  12  miles? 

Ans.  4752. 

18.  If  8  horses  eat  12  bu.  3  pk.  of  oats  in  3  days,  how 
many  bushels  will  20  horses  eat  in  the  same  time  ? 

Ans.  31  bu.  3  pk.  4  qt. 

19.  How  much  sugar  at  9  cents   a   pound  must  be  given 
for  2  cwt.  43  lb.  of  pork  at  6  cents  a  pound  ? 

Ans.  162  pounds. 

20.  How  many  cubic  feet  in  a  room  18  feet  long,  16  feet 
wide,  and  10  feet  high  ? 

21.  A  person  wishes   to  ship  720  bushels  of  potatoes  in 
barrels,  which  shall   hold  3  bu.  3  pk.  each,  how  many  bar- 
rels must  he  use  ?  Ans.  192. 

22.  How  many  rods  of  fence  will   inclose  a  farm  a  mile 
square  ?  Ans.  1280  rods. 

23.  If  granite  weigh    175  pounds   a  cubic  foot,  what   ia 
the  weight  of  a  cubic  yard  ?          Ans.  2  T.  7  cwt.  25  lb. 


CANCELLATION.  167 


CANCELLATION. 

174:.  Cancellation  is  the  process  of  rejecting  equal 
factors  from  numbers  sustaining  to  each  other  the  relation 
of  dividend  and  divisor. 

It  has  been  shown  (  70  )  that  the  dividend  is  equal  to 
the  product  of  the  divisor  multiplied  by  the  quotient. 
Hence,  if  the  dividend  can  be  resolved  into  two  factors, 
one  of  which  is  the  divisor,  the  other  factor  will  be  the 
quotient. 

1.  Divide  72  by  9. 

OPERATION.  ANALYSIS.     We  see  in  this 

Divisor.  0)0  X$  Dividend.         example,  that  72  is  composed 

-  of  the  factors  9  and  8,  and 

8   Quotient.         that  the  factor  9,  is  equal  to 

the  divisor.     Therefore  we  reject  the  factor  9,  and  the  remain- 

ing factor,  8,  is  the  quotient. 

174.  Whenever  the  dividend  and  divisor  are  each 
composite  numbers,  the  factors  common  to  both  may  first 
be  rejected  without  altering  the  final  result. 

2.  What  is  the  quotient  of  48  divided  by  24  ? 

OPERATION.  ANALYSTS.      We   first  indi- 

48     $X$X2  wte  the  operation  to  be  per- 

-^-:=~2  Ant.  "  IbVmed;  by  wfrtiifg  the  dividend 
above  a  line,  and  the  divisor 


below  it.  We  resolve  48,  into  the  factors  3,  8  and  2,  and  24  in- 
to the  factors  3,  and  8.  We  next  cancel  the  factors  3,  and  8, 
which  are  common  to  the  dividend  and  divisor,  and  we  have 
left  the  factor  2,  in  the  dividend,  which  is  the  quotient. 

NOTE.    When  all  the  factors  or  numbers  in  the  dividend  are  cancelled,  1  should 
bo  retained. 


163  CANCELLATION. 


If  any  two  numbers,  one  in  the  dividend  and  one 
in  the  divisor,  contain  a  common  factor,  we  may  reject 
that  factor.  ^ 

3.  In  15  times  63,  how  many  t'imes  457 

OPERATION.  ANALYSIS.     In  this  example  we  see 

that  5  will  divide  15  and  45  ;  so  we 
Ans  reJec*  5  as  a  factor  of  15,  and  retain 
the  factor  3,  and  also  as  a  factor  of  45, 
and  retain  the  factor  9.  Again  9  will 
divide  9  in  the  divisor,  and  63  in  the 

dividend.  Dividing  both  numbers  by  9,  1  will  be  retained  in 
the  divisor,  and  7  in  the  dividend.  Finally  the  product  of  3  X 
7  =  21,  the  quotient. 

4.  What  is  the  quotient  of  25x18X^X4,  divided  by 
15X4X9X3? 

OPERATION. 

2  ANALYSIS.  In 

O  this,  as  in  the 


3  3  preceedingez- 

s  ample,  we  re- 

ject all  the  factors  that  are  common  to*  both  dividend  and 
divisor,  and  we  have  remaining  the  facjtoi^S^a.^^Hiiyjs^.  and 
the  factors  5,  2,  and  2  in  the  dividend.  Completing  the  work, 
we  have  23°=6|,  Av$.  ^ 

From  the  precee^jfog^examples  and  illustrations  we  de- 
rive the  following    :  • 

BuLTT-T-^Brff.  ^^^^^^f^tay^-mw^r 

above  a  horizontal  line,  and  the  numbers  composing  the  di- 
visor below  it. 

II.  Cancel  all  the  factors  common  to  both  dividend  and 
divisor. 

III.  Divide  the  product  of  the  remaining  factors  of  the 
dividend  by  the  product  of  the  remaining  factors  of  the  di- 
visor ,  and  the  result  will  be  the  quotient. 


CANCELLATION. 


169 


Nones.     Ir  Rejecting  a  factor  from  any  number  is  dividing  the  numb«r  by  that 
factor. 
2-  When  a  factor  is  cancelled,  the  unit,  1,  is  supposed  to  take  its  place. 

3.  One  factor  in  the  dividend  will  cancel  only  one  equal  factor  hi  the  diviaor. 

4.  If  all  the  factors  or  numbers  of  the  divisor  are  cancelled,  the  product  of  th« 
remaining  factors  of  the  dividend  will  be  the  quotient. 

5.  By  many  it  is  thought  more  convenient  to  write  the  factors  of  the  dividend  on 
the  right  of  a  vertical  line,  and  the  factors  of  the  divisor  on  the  left. 

EXAMPLES  FOR  PRACTICE. 

1.  Divide  the  product  of  12x8x6  by  8x4X3. 


FIRST   OPERATION. 


3X2 


-=6  Ans. 


SECOND   OPERATION. 

If 


6  Ans. 


2.  Divide  the  product  of  25x18x4x3, 

FIRST   OPERATION. 

&X20X4XJ     5X3X4    60 


04 


\  - 


84  Ans. 
3.  Divide  the  uroduct  of  36x10X7  by  14x5x9. 


.    4. 

4.  What  is  the  quotient  of  21X8X40X3  divided  by 
12X7X20?  Ans.  12. 

5.  What  is  the  quotient  of  64x18x9  divided  by  30  X 
27X4?  Ans.  3f 

6.  Divide  the  product  of  120x44x6  by  60x11X8. 

Ans.  6 
8 


170  CANCELLATION. 

7.  Multiply  200  by  60,  and  divide  the  product  by  50 
multiplied  by  48.  Ans.  5. 

8.  Multiply  8  times  32  by  6  times  27,  and  divide  the 
product  by  9  times  96.  Ans.  48. 

9.  What  is  the  quotient  of  21x8x60x8x6  divided  by 
7X12X3X8X3?  Ans.  80. 

10.  What  is  the  quotient  of  18x6x4x42  divided  by 
4X9X3X7X6?  Ans.  4. 

11.  If  18X5X^X66  be  divided  by  40x22x6,  what  is 
the  quotient?  Ans.  10^. 

12.  The  product  of  thj  numbers  26,  11,  and  21,  is  to  be 
divided  by  the  product  of  the  numbers  14  and  13 ;  what  is 
the  quotient  ?  Ans.  33. 

13.  The  product  of  the  numbers  48,  72,  28  and  5,  is  to  be 
divided  by  the  product  of  the  numbers  84,  15,  7  and  6; 
what  is  the  quotient  ?  Ans.  9^. 

14.  How  many  tons  of  hay  at  $9  a  ton,  must  be  given  for 
27  cords  of  wood,  at  $4  a  cord  ?  Ans.  12  tons. 

15.  How  many  bushels  of  corn,  worth  60  cents  a  bushel, 
must  be  given  for  25  bushels  of  rye,  worth  90  cents  a 
bushel?  Ans.  37^  bushele. 

16.  How  many  peaches  worth  2  cents  eaoh  must  be  given 
for  48  oranges,  at  3  cents*  each  ?  Ans.  72 

17.  How  many  days  work,  at  75  cents  a  day,  will  pay  for 
30  pounds  of  coffee,  at  15  cents  a  pound  ?     Ans.  6  days. 

18.  How  many,  suits  of  clothes,  at  $18  a  suit,  can  be  made 
from  5  pieces  of  cloth,  each  piece  containing  24  yards,  at 
$3  a  yard  ?  Ans.  20  suits. 

19.  How  many  tubs  of  butter,  each  containing  48  pounds, 
at  14  cents  a  pound,  must  be  given  for  3  boxes  of  tea,  each 
containing  42  pounds,  worth  60  cents  a  pound  ? 

Ant.  ll 


CANCELLATION.  171 

20.  How  many  days  work,  at  84  cents  a  day,  will  pay 
for  36  bushels  of  corn  worth  56  cents  a  bushel1? 

Ans.  24. 

21.  A  farmer  exchanged  45  bushels  of  potatoes  worth  30 
cents  a  bushel,  for  15   pounds  of  tea;  what  was  the  tea 
worth  a  pound?  Ans.  90  cents. 

22.  A  grocer  bought  120  pounds  of  cheese,  at  9  cents  a 
pound,   and  paid  in  molasses,  at  45  cents  a  gallon ;  how 
many  gallons  of  molasses  paid  for  the  cheese  1 

Ans.  24  gallons. 

23.  Gave   12  barrels  of  flour,  at  $7  a  barrel,  for  hay 
worth  818  a  ton ;  how  many  tons  of  hay  was  the  flour 
worth?  Ans.  4§  tons. 

24.  Sold  8  firkins  of  butter,  each  weighing  56  pounds, 
at  15  cents  a  pound,  and  received  in  payment  3  boxes  of 
tea,  each  containing  40  pounds;  how  much  was  the  tea 
worth  a  pound  ?  Ans.  56  cents. 

25.  A  man  took  6  loads  of  apples  to  market,  each  load 
containing  14  barrels,  and  each  barrel  3  bushels.     He  sold 
them  at  50  cents  a  bushel,  and  received  in  payment  9  bar- 
rels of  sugar,  eac^.  weighing  210  pounds ;  how  much  was 
the  sugar  worth  a  pound  1  An-s.  6|  cents. 

26.  A  grocer  sold  12  boxes  of  soap,  each  containing  51 
pounds,  at  10   cents  a  pound ;  he  received  in  payment  a 
certain  number  of  barrels  of  potatoes,  each  containing  3 
bushels,  at  30  cents  a  bushel ;  how  many  barrels  did  he 
receive  ?  Ans.  68  barrels. 

27.  A  man  sold  4  loads  of  barley,  each  load  containing 
60  bushels,  at  70  cents  a  bushel,  and  received  in  payment 
2  pieces  of  cloth,  each  piece  containing  35  yards,  how  much 
was  the  cloth  worth  a  yard  ?  -4ns.  $2.40. 


172  ANALYSIS. 


ANALYSIS. 

176.  Analysis,  in  arithmetic,  is  the  process  ot  solving 
problems  independently  of  set  rules,  by  tracing  the  relations 
of  the  given  numbers  and  the  reasons  of  the  separate  steps 
of  'the   operation   according  to  the  special   conditions   of 
each  question. 

177.  In  solving  questions  by  analysis,  we  generally  rea- 
son  from   the  given  number   to  unity,  or  1;  and   then  from 
unity,  or  1,  to  the  required  number. 

178.  United  States  money  is  reckoned  in  dollars,  dimes, 
cents,  and  mills,  one  dollar  being   uniformly  valued   in  all 
the  States  at  100  cents ;  but  in  most  of  the  States  money  is 
sometimes  still  reckoned  in  pounds,  shillings  and  pence. 

NOTB.  At  the  time  of  the  adoption  of  our  decimal  currency  by  Congress,  in 
1786,  the  colonial  currency,  or  bills  of  credit,  issued  by  the  colonies,  had  depreciated 
in  value  and  this  depreciation,  being  unequal  in  the  different  colonies,  gave  rise  to 
the  different  values  of  the  State  currencies  ;  and  this  variation  continues  wherever 
the  denomination  of  shillings  and  pence  are  in  use, 

Georgia  Currency. 
Georgia;  South  Carolina, $l=4s.  8d.—  56d. 

Canada  Currency. 

Canada,  Nova  S,cotia, $l=»5s.=60d. 

New  England  Currency. 

NewEnglanl  Sta  es,  Indiana,  Illinois,  J 

Missouri,  Virginia,  Kentucky,   Tennes-> $1— 6s.— 72d. 

see,  Mississippi,  Texas, ) 

Pennsylvania  Currency. 
New  Jersey,  Pennsylvania,  Delaware, )  ^  ^  $l_7s  6d.— 90d. 

New  York  Currency. 

New  York,  Ohio,  Michigan,  )  *,     fta     or , 

North  Carolina, f *] 

In  many  of  the  States  it  is  customary  to  give  the  retail  price 
of  articles  in  shillings  and  pence,  and  the  cost  of  the  whole  in 
dollars  and  cents. 


ANALYSIS.  173 

The  following  will  be  found  an  easy,  shoit,  and  practical 
method  of  reducing  currencies  to  dollars  and  cents. 

EXAMPLES   FOR   PRACTICE. 

1.  What  will  be  the  cost  of  36  bushels  of  apples,  at  3 
shillings  a  bushel,  New  England  Currency  ? 

OPERATION.  ANALYSIS. Since    1 

6  bushel  costs  3  shillings, 

36X3  =  108s.           I  $0  3G  bushels  will  cost  36 

108^-6  =  $18    Or  0  I  3  times3s.,or36x3~l08s.; 

~~  and  as  6s.  make  1  doUar, 

18,  Ans.     New  Engknd   cuvrencV| 

there  are  as  many  dollars  in  108s.  as  6  is  contained  times  in  108, 
or  108-r-6=i 


2.  What  will  112  bushels  of  barley  cost,  at  5s.  6d.  per 
bushel,  New  York  currency  ? 


OPERATION. 


7 


Or 


XX  ft 


*  ANALYSIS.  —  We  mul- 


U  tiply    the    number    of 

bushels   by   the   price, 

$77  and  divide  the  result  by 

$77   Ans.     the  value  of  1  dollar  as 

in  the  first  example,  reducing  both  the  price  and  1  dollar  to  pence, 
and  we  obtain  $77.  Or,  when  the  price  is  an  aliquot  part  of  a 
shilling,  the  price  may  be  reduced  to  an  improper  fraction  for  a 
multiplier,  thus;  5s.  6d—  5 As.  —3-8.,  the  multiplier.  The  value 
of  a  dollar  being  8s.,  we  divide  by  8  as  in  the  operation.  Hence 
To  find  the  cost  of  articles  in  dollars  and  cents,  when  the  price 
is  in  shillings  and  pence, 

Multiply  the  commodity  by  the  price,  and  divide  the  pro- 
duct by  the  value  of  one  dollar  in  the  required  currency, 
reduced  to  the  same  denominational  unit  as  the  price. 


174  ANALYSIS. 

3.  What  will  180  cords  of  wood  cost  at  8s.  4d.  per  cord, 
Pennsylvania  currency? 

OFEKATION.  ANALYSIS. — Multiply 


2 


Or, 


100 


$200 


4  the    quantity    by    the 


price  in  pence,  and*  di- 
vide the  product  by  the 
value  of  1  dollar  in 


$200,  Ans.  ,        ., 

pence ;    or,  reduce  the 

shillings  and  pence,  both  of  the  price  and  of  the  dollar,  to  the 
fraction  of  a  shilling  before  multiplying  and  dividing,  thus ; 
8s.  4d.=8^s.  "=235s>»  ^ne  multipner'  The  value  of  the  dollar 
being  7s.  6d.  =7£s.  —  ^s.  we  divide  by  ^  as  in  the  operation. 

4.  What  will  be  the  cost  of  7  J  yards  of  cloth,  at  6s.  8d. 
New  York  currency  ? 

OPERATION. 


,  -  or  AA  ANALYSIS. — We  reduce  the  quan- 

'        tity  and  the  price  to  improper  frao 

$6.25     Ans.       tions>  before  multiplying. 

NOTE.— When  there  is  a  remainder  in  the  dividend,  it  may  be  reduced  to  cents 
and  mills  by  annexing  two  or  three  ciphers  and  continuing  the  division. 

5.  What  will  7  hhd.  of  molasses  cost  at  Is.  3d.  per  quart, 
Georgia  currency1? 

OPERATION.  ANALYSIS. — In    this   example   we 

%  first  reduce  7  hhd.  to  quarts,  by  mul- 

63  tiplying  by  63,  and  4,  and  then  mul- 

^  tiply  by  the  price,  either  reduced  to 

pence  or  to  an  improper  fraction,  and 
2  I  Q45  00  divide  by  the  value  of  1  dollar  re 

'  duced  to  the  same  denomination  aa 

$472.50  Ans.     the  price. 


ANALYSIS.  175 

6.  Sold  8  firkins  of  butter,  each  containing  56  pounds  at 
Is.  3d.  per  pound,  and   received   in   payment  tea  at  6s.  8d. 
per  pound;  how  many  pounds  of  tea  would  pay  for  the  butter? 

OPERATION.  ANALYSIS. — The    operation    in 

28  this  is  similar  to   the   preceding 

03  examples,  except  that  we  divide 

9  the  cost  of  the  butter  by  the  price 

of  a  unit  of  the  article  received  in 
Ans.  84  pounds.  ,  ,  ,  ., 

payment,  reduced  to  the  same  de- 
nominational unit  as  the  price  of  a  unit  of  the  article  sold.  The 
result  will  be  the  same  in  whatever  currency. 

7.  What  will  be  the  cost  of  a  load  of  oats  containing  64 
bushels  at  2s.  6d.  a  bushel,  New  York  currency  ? 

Ans.  $20. 

8.  At  9d.  a  pound,  what   will  be  the  cost  of  120  pounds 
of  sugar,  New  England  currency?  Ans.  £15. 

9.  What  will   be  the  value  of  a   load  of  potatoes,  meas- 
uring 35  bushels,  at  2s.  3d.  a  bushel,  Penn.  currency? 

Ans.  $10.50. 

10.  What  will   be   the  cost  of  240   bushels  of  wheat,  at 
9s.  4d.  a  bushel,  Michigan  currency  1  Ans.  &2SO. 

11.  In  New  Jersey  currency  ?  Ans.  $298.66|. 

12.  In  IHinois  currency?  Ans.  $373.33^. 

13.  In  South  Carolina  currency  ?  Ans.  $480. 

14.  In  Virginia  currency  ?  Ans. 

15.  In  Ohio  currency  ?  Ans. 

16.  In  Canada  currency  ?  Ans.  $448. 

17.  How  many  days  work  at  7s.  6d.  a  day,  must  be  given 
for  5  bushels  of  wheat  at  10s.  a  bushel  ?     Ans.  6f  days. 

18.  What  will  be  the  cost  of  5  casks  of  rice,  each  weigh- 
ing 168  pounds,  at  3d.  per  pound,  South  Carolina  currency  1 

Ans.  $45. 


176  ANALYSIS. 

19.  How  many  pounds  of  sugar^at  9d.  per  pound,  must 
be  given  for  18  bushels  of  apples,  at  2s.  7d.  per  bushel  ? 

Ans.  62  pounds. 

20.  Bought  3  casks  of  catawba  wine,  each  cask  contain- 
ing 64  gallons,  at  7s.  9d.  per  quart,  Ohio  currency ;  what 
was  the  cost  of  the  whole  1  Ans.  $744. 

21.  What  will  it  cost  to  build  150  rods  of  wall,  at  3s.  8d. 
per  rod,  Canada  currency  ?  Ans.  $110. 

22.  How  many  pounds  of  butter,  at  18d.  a  pound,  must 
be  given  for  12  pounds  of  tea,  at  5s.  4d.  a  pound  1 

Ans.  42|  pounds. 

23.  What  will  be  the  cost  of  4  hogsheads  of  molasses,  at 
Is.  2d.  per  quart,  Mississippi  currency  1          Ans.  $196. 

24.  A  farmer  exchanged  28  bushels  of  barley/  worth  5s. 
8d.  a  bushel,  with  his  neighbor,  for  corn  worth  7s.  a  bushel; 
how  many  bushels  of  corn  was  the  barley  worth  ? 

Ans.  22|  bushels. 

25.  What  will  a  load  of  wheat,  measuring  45  bushels,  be 
worth  at  lls.  a  bushel,  Kentucky  currency  ? 

Ans.  $82.50. 

26.  What  will  12  yards  of  Irish  linen  cost,  at  4s.  9d.  a 
yard,  Pennsylvania  currency?  J.TI&.  $7.60. 

27.  Bought  the  following  bill  of  goodg  of  f  radewell  & 
Co. ;  how  much  did  the  whole  amount  to,  New  York  cur- 
rency ? 

4  yards  of  cloth  at         5s.  6d.  per  yard, 

9     "          calico,  -         -    "    -     Is.  4d.         « 
10     "          ribbon,     -         «         2s.  3d.         « 
6  gallons  molasses,        -    "         4s.  8d.  per  gallon,  " 
3  J  pounds  of  tea,      -         "         6s.  per  pound. 

Ans.  $13.1875. 


PERCENTAGE.  177 


PERCENTAGE. 

179.  Per  cent  is  a  term  derived  from  the  Latin  words 
per  centum,  and  signifies  by  the  hundred,  or  hundredths,  that 
is,  a  certain  number  of  parts  of  each  one  hundred  parts,  of 
whatever  denomination.  Thus,  by  5  per  cent,  is  meant  5 
cents  of  every  100  cents,  $5  of  every  $100,  5  bushels  of 
every  100  bushels,  &c.  Therefore,  5  per  cent,  equals  5 
hundredths^.OS^jf^T^.  8  per  cent,  equals  8  hun- 
dredths =  .08=T$  0 = 22~. 

1 8O.  Percentage  is  such  a  part  of  a  number  as  is  in- 
dicated by  the  per  cent. 

181.  The  Base  of  percentage  is  the  number  on  which 
the  percentage  is  computed. 

1 82.  Since  per  cent,  is  any  number  of  hundredths,  it 
is  usually  expressed  in  the  form  of  a  decimal. or  a  common 
fraction,  as  in  the  following 

*       TABLE. 
Decimals.        Common  Fractions.      Lowest  Term* 

1  per  cent        —        .01        —        T^        —        T^ 

2  per  cent         "        .02         "         T3s         "          A 

4  per  cent  "  .04  "  ^  ^ 

5  per  cent  "  .05  "  T«^  J^ 

6  per  cent.  ••  ,.06  "  Tfo  *  ^ 

7  per  cent  "  07  "  T^  "  jfa 

8  per  cent.  "  .08  "  Tfo  "  & 
10  per  cent.  «  .10  "  J^  «  ft 
16  percent  "  .16  «  ^  "  2\ 
20  per  cent  «'  .20  "  ^  ««  J 
25  per  cent  "  .25  "  ^  a  ^ 
50  per  cent.  '«  .50  "  T»&  "  % 

100  per  cent.         "      1.00         "         |jj  '      "  I 


178  PERCENTAGE. 


.  To  find  the  jfig^centage  of  any  number. 

1.  A   man  having   $12*0,  paid  out  5  per  gent,  of  it  for 
groceries  ;  how  much  did  he  pay  out  1 

OPERATION. 
$120 
.05 

_  ANALYSIS.  —  Since  5  per  cent,  is  T5  o  """ 

$6.00  .05,  he  paid  out  .05  of  $120,  or  $120X05 

=$6.     Hence  the 

RULE.  Multiply  the  given  number  or  quantity  by  the 
rate  per  cent,  expressed  decimally,  and  point  off  as  in  dec- 
imals. 

EXAMPLES   FOR    PRACTICE. 

2.  What  is  4  per  cent,  of  $300  ?  Ans.  $12. 

3.  What  is  3  per  cent,  of  $175?  Ans.  $5.25. 

4.  What  is  5  per  cent,  of  450  pounds  ? 

5.  What  is  6  per  cent,  of  65  gallons  ?  Ans.  3.9  gal. 

6.  What  is  9  per  cent,  of  200  sheep  ?  Ans.  18  sheep. 

7.  What  is  7  per  cent,  of  $97?  Ans.  $6.79. 

8.  What  is  10  per  cent,  of  $12.50  ?  Ans.  $1.25. 

9.  What  is  40  per  cent,  of  840  men?  Ans.  336  men. 

10.  What  is  25  per  cent,  of  740  miles  ? 

11.  A  man  having   $4000,  invests  25  per   cent,  of  it   in 
land;  what  sum  does  he  invest?  Ans.  $1000. 

12.  A  man   bought   1500  barrels  of  apples,  and  found  on 
opening  them  that  12    per  cent,  of  them  were  spoiled  ;  how 
many  barrels  did  he  lose  ?  Ans.  180  barrels. 

13.  A  farmer  having  180  sheep,  sold  45  per  cent,  of  them 
and   kept   the  remainder;  how   many  did  he  sell  and  how 
many  did  he  keep  1  Ans.  He  kept  99. 

14.  Having  deposited  $1275  in   bank,  I  draw  out  8  per 
cent,  of  it;  how  much  remains?  Ans.  $1173. 


COMMISSION.  179 


COMMISSION. 

1 84.  An  Agent,  Factor,  or  Broker,  is  a  person  who 
transacts  business  for  another. 

1  §5.  A  Commission  Merchant  is  an  agent  who  buya 
and  sells  goods  for  another. 

186.  Commission  is  the  fee  or   compensation  of  an 
agent,  factor,  or  commission  merchant. 

187.  To  find  the  commission  or  brokerage  on  any 
sum  of  money. 

1.  A  commission  merchant  sells  butter  and  cheese  to  the 
amount  of  $1540 ;  what  is  his  commission  at  5  per  cent.  ? 

OPERATION.  *  ANALYSIS.     Since  the  com- 

$1540X-05=$77,  Ans.         mission  on  $1  is  5  cents  or 
.05  of  a  dollar,  on  $1540  it  is  $1540X.05«=$77.     Hence  the 

RULE.  Multiply  the  given  sum  by  the  rate  per  cent, 
expressed  decimally  ;  the  result  will  be  the  commission  or 
brokerage. 

EXAMPLES   FOR    PRACTICE. 

2.  What  commission  must  be  paid  for   collecting  $3840, 
at  3  per  cent.  ?  Ans.  $115.20. 

3.  A  commission  merchant   sells  goods    to  the  amount  of 
$5487.50;  what  is  his  commission,  at  2  per  cent.  ? 

Ans.  $109.75. 

4.  An   agent  buys   5460  bushels  of  wheat   at   $1.50   a 
bushel ;  how  much   is   his  commission  for   buying,  at  4  per 
cent.?  Am.  $327.60. 

5.  A  commission  merchant  sells  400  barrels  of  potatoes 
at  $2.25  a  barrel,  and  345  barrels  of  apples  at  $3.20  a  bar 
rel ;  how  much  is  his  commission  for  selling,  at  5  per  cent.  ? 

6.  An  age/nt  sold  my  house  and  lot  for  $6525 ;  what  wasi 
his  commission  at  2  per  cent.  ] 


180  PERCENTAGE. 

"Y*  PROFIT  AND  LOSS. 

18 8. "Profit  and  Loss  are  commercial  terms,  used  to 
express  the  gain  or  loss  in  business  transactions,  which  is 
usually  reckoned  at  a  certain  per  cent,  on  the  prime  or  first 
cost  of  articles. 

1 89.  To  find  the  amount  of  profit  or  loss,  when 
the  cost  and  the  gain  or  loss  per  cent,  are  given. 

1.  A  man  bought  a  horse  for  $135,  and  afterward  sold 
him  for  20  per  cent,  more  than  he  gave ;  how  much  did  he 
gain? 

OPERATION. 

ANAWESIS. — Since  $1   gains   20 

$135  X-20^2?,  ^ns-      cents,  or  20  per  cent.,  $135  will 
gain  $135X-20=$27.     Hence  the 

RULE.  Multiply  the  cost  by  the  rate  per  cent,  expressed 
decimally-. 

EXAMPLES   FOR   PRACTICE. 

2.  Bought  a  horse  for  $150,  and  sold  him  at  15  per 
cent,  profit;  how  much  was  my  gain?  Ans.  $22.50. 

3.  Bought  25  cords  of  wood  at  $3.50  a  cord,  and  sold 
it  so  as  to  gain  33  per  cent. ;  how  much  did  I  make  ? 

Ans.  $28.87*. 

4.  Paid  7  centra  pound  for  2480  pounds  of  pork,  and 
afterward    lost  10  per  cent,  on  the  cost,  in  selling  it ;  how 
much  was  my  whole  loss  ?  Ans.  $17.36. 

5.  Bought  1000  bushels  of  wheat  at  $1.25  a  bushel, 
and  sold  the  flour  at  18  per  cent,  advance  on  the  cost  of  the 
wheat ;  how  much  was  my  whole  gain  ?  Ans.  $225. 

6.  A  grocer  bought  6  barrels  of  sugar,  each  containing 
220  pounds,  at  7J  cents  a  pound,  and  sold  it  at  20  per  cent 
profit;  how  much  was  the  whole  gain  ?        Ans.  $19.80. 


SIMPLE   INTEREST. 


181 


SIMPLE  INTEREST. 

190.  Interest  is  a  sum  paid  for  the  use  of  money. 

191.  Principal  is  the  sum  for  the  use  of  which  in- 
terest is  paid. 

1 92.  Rate  per  cent,  per  annnm  is  the  sum  per  cent, 
paid  for  the  use  of  $100  annually. 

NOTE.    The  rate  per  cent,  is  commonly  expressed  decimally,  as  hnndredtha. 
(182.) 

193.  Amount  is  the  sum  of  the   principal   and  in- 
terest. 

194.  Simple  Interest  is  the  sum   paid   for   the  use 
of  the  principal  only,  during  the  whole  time  of  the  loan 
or  credit. 

195.  Legal  Interest  is  the  rate  per  cent,  establish- 
ed by  law.     It  varies  in  different  States,  as  follows : 

Minnesota, 'jS)per  cent. 

Mississippi, (3. 

Missouri, .6 

New  Hampshire,.  .6 

New  Jersey, 6 

New  York, ft/ 

North  Carolina, ...  6 

Ohio, 6 

Pennsylvania,  ....  6 
Rhode  Island  ....  6 
South  Carolina,...; 7 


Alabama,  .....  .'.  .  .8  per  cent. 

Arkansas,  ........  6 

California,.  ......  10 

Connecticut,  ......  6 

'Delaware,  ........  6 

Dist  of  Columbia,.  6 
Florida,  ..........  8 

......  7 

......  .6 

Indiana,  .........  6 

Iowa,  ............  7 

Kentucky,  .......  6 

Louisiana  ........  5 

Maine.  .  ;  .....  .  ...6 

Maryland,  ........  6 

Massachusetts,  ....  6 

Michigan,  ........  7 


Tennessee,  .......  6 

Texas,  ..........  .8  ', 

U.S.  (debts),....^.  " 

Vermont,  .......  ..6/  " 

Virginia,  .......  jfrf    « 

\F  " 


Wisconsin, 


NOTES.  1.  The  legal  rate  in  Canada,  Nova  Scotia,  and  Ireland  is  6  per  cent.,  and 
In  England  and  France  5  per  cent. 

2.  When  the  rate  per  cent,  is  not  specified  in  accounts,  notes,  mortgages,  con- 
tracts, &c.,  the  legal  rate  is  always  understood. 


182  PERCENTAGE. 

CASE   I. 

196.  To  find  the  interest  on  any  sum,  at  any  rate 
per  cent.,  for  years  and  months. 

1.  What  is  the  interest  on  $140  for  3  years  3  months, 
at  7  per  cent.  ? 

OPERATION. 

$140 

.07  ANALYSIS.  —  The  interest  on 

$140,  for  1  yr.,  at  7  per  cent, 

$9.80  int.  for  1  year.         [a  .07  of  the  principal,  Or  $9.- 

_  _^  80,  and*?h%  interest  for  3  yr. 

245  3  mo.  is  3T%—3|  times  the 

2940  interest  for  one  yr.,  or^$9.80 

X  3  J,  which  is  $31. 
Ans.    $31.85  Int.  for  3  yr.  3  mo.     Hence,  the  following 


RULE.  I.  Multiply  the  principal  fylthe  rate  per  cent.t 
and  the  product  will  be  the  interest  for  1  year. 

II.  Multiply  this  product  fy  the  time^in  years  and  frac- 
tions of  a  year,  and  the  result  will  be  the  required  interest. 

EXAMPLES   FOR  PRACTICE. 

2.  What  is  the  interest  on  $48.50  for  2  years  6  months, 
at  6  per  cent.  ?  Ans.  $7x275. 

3.  What   is   the   interest   on    $325.41    for   3  years.! 
months,  at  5  per  cent.  1  Ans.  $54.235. 

4.  What  is  the  interest  on  $279.60  for  1  year  9  months, 
at  7  per  cent.  7  -  Ans.  $34.251. 

5.  What  is  the  amount  of  $26.84  for  2  yr.  6  mo.,  at  5 
per  cent.  1  Ans.  $30.195. 

6.  What  is  the  amount  of  $200  for  1  yr.  9  mo.,  at  7 
percent?  Ans.  $224.50. 

7.  What  is  the  interest  on  $750  for  1  year^3  months, 
at  5  per  cent.  1  Ans.  $46.875. 


SIMPLE  INTEREST.  183 

CASE   II. 

197.  To  find  the  interest  on  any  sum,  for  any 
time,  at  any  rate  per  ccjnt. 

Obvious  Relations  between   Time  aud  Interest. 

I.  The  interest  on  any  sum  for  1  year,  at  1  per  cent.,  is 
.01  of  that  sum,  and  is  equal  to  the  principal  with  the  sep- 
eratrix  removed  two  places  to  the  left. 

II.  A  month  being  -^  of  a  year,  J2  of  the  interest  on 
any  sum  for  1  year  is  the  interest  for  1  month. 

III.  The  interest  on  any  sum  for  3  days  is  ^=-^=.1 
of  the  interest  for  1  month,  and  any  number  of  days  may 
readily  be  reduced  to  tenths  of  a  month  by  dividing  by  3. 

IV.  The  interest  on  any  sum  for  1  month,  multiplied 
by  any  given  time  expressed  in  months  and  tenths  of  a 
month,  will  produce  the  required  interest. 

^\.  What  is  the  interest  on  $306  for  1  yr.  6  mo.  12  da., 
at  7  per  cent  ? 

OPERATION.  ANALYSIS.     Removing   the 

]  p*.  6  mo.  12  da.  =  18.4  mo.  seperatrix  in  the  given  princi- 

12)$3.060  Pal    two  places    to  the   left, 

we  have  $3. 06,  the  interest  on 

$.255  the  given  sum  for  1  year  at  1 

per  cent.     (I).  Dividing   this 

— —  by  12,  we  have  $.255,  the  inter- 

2040  GSt  f°r  *  montll>  at  !  Per  cent* 

(!!)•    Multiplying    this    quo- 
tient by   18.4,   the  time    ex- 
pressed  in  months   and  deci- 
mals of  a  month.     (HI),  we 
$32.8440  Ans.        have  $4.692,  the  interest  on  the 
given  sum  for  the  given  time,  at  1  per  cent.     (IV).  And  multi- 
plying this  product  by  7,  the  rate  per  cent,  we  have  $32.844,  the 
required  interest.     Hence, 


VUl 

1 


184  PERCENTAGE. 

RULE.  I.  Remove  the  separatrix  in  the  given  principal 
two  places  to  the  left ;  the  result  will  be  the  interest  f  or  \ 
year  at  1  per  cent. 

II.  Divide  this  interest  by  12  f  the  result   will   be  the  in- 
terest for  1  month,  at  1  per  cent. 

III.  Multiply  this  interest  by  the  given  time  expressed  in 
months  and  tenths  of  a  month  ;  the  result  will  be  the  interest 
for  the  given  time,  at  1  per  cent. 

IV.  Multiply  this  interest  by  the  given  rate  ;  the  product 
wiU  be  the  interest  required. 

EXAMPLES  FOR  PRACTICE. 

2.  What  is  the  interest  on  $34.25  for  3  yr.  8  mo.  15  da., 
5  per  cent.  ?  Ans.  $6.35. 

3.  What  is  the  interest  on  $260  for  9  mo.  <f  da.,  at  6  per 
cent.?  Ans.  $11.826, 

4.  What  is  the  interest  on  $450,  at  6  per  cent,  for   10 
mo.  18  days?  Ans.  $23.85. 

5.  What  is  the  interest  on  $372  for  1  yr.  10  mo.  15  days, 
at  7  per  cent.  ?  Ans.  $48.825. 

6.  What  is  the  interest  on  $221.75  for  3  yr.  7  mo.  6  da., 
at  7  per  cent.  ?  .  Ans.  $55.88. 

8.  What  is  the  interest  on  $267.27  for  6  mo.  24  days,  at 

6  per  cent.  ?  Ans.  $9.086. 

9.  What  is  the   interest  on   $365  for  2  mo.  3  days,  at  6 
per  cent.  ?  Ans  .$3.83. 

10.  What  is  the  interest  on   $785.10,  for  1  yr.  6  months 
18  days,  at  5  per  cent.  ?  Ans.  $60.845. 

11.  On  $450  for  3  yr.  7  months,  at  8  per  cent.? 

2.  What   is   the  interest  on   $600  for  2  yr.  8  mo.,  at  7 
r  cent.  ?        *  Ans.  $112. 

13.  What  is  the  amount  of  $1000  for  9  mo.  15  days,  at 

7  per  cent.?  Ans.  $1055.414. 


INTEREST.  186 

14.  What  is  the  interest  on  $860  for  6  mo.  6  days,,  at  6 
per  cent.  ?        *  i    -x  l|  •  4  A  .^ras.  $26.66. 

15.  What  istthe  interest  on  $137.45  for   8  mo.  27  days, 
at  6  per  cent.  ?  I 

^^J.6.  Find   the   amount  of  $8 Jo  for  1   yr.  6  mo.  at  3  per 
"'cent.'?  Ans.  $914.375. 

v!7.  Find  the  amdfint  of  $350vfor  9  mo.,  at  4  per  cent  1 
\  %  X      Ans.  $360.497. 

18.  Find'the  amount  o|%8.50  for  1  yr.  9  mo.  12  da.,  at 
6  per  cent.  ?  Ans.  $9.409. 

19.  Find  the  amount  of  $457  for  1  yr.  4  mo.  24  da.,  at 

6  per  cent.  ?  .  Ans.  $495.388. 

20.  Find  the  amount  of  $650  for  3  yr.  10  mo.  21  days  at 

7  per  cent.  ?  ^  Ans.  $827.009. 

21.  What  is  the  interest  on   $79  for   15   mo.,  at  7   pei 
cent.  ?  Ans.  $6.912  x  . 

22.  Find  the  amount  of  $.86  for  5  mo.,  7  per  cent. 

V  Ans.  $.885. 

23.  What  is  the  interest  on  $78.75  for   1  yr.  9  mo.,  at  4 
per  cent.  ?  Ans.  $5.5125. 

24.  What  is  the  interest  on  $1750  for  30  days,  at  9  per 
cent.  ?  Ans.  $13.125. 

25.  What  is  the««terest  on  $3654.25  for  33  days,  at  10 
per  cent.  ?  Ans.  $33.497. 

26.  Find  the  amount  of  $269.50  for  120  days,  at  7  pei 
cent.?  Ans.  $275.788. 

7.  Find  the  amount  of  $1625  for  1  yr.  6  mo.,  at   8  per 
t.1  Ans.  $1820. 

NOTE. — For  a  full  treatise  of  Percentage  in  all  its  applications  to  the  businesi 
transactions  of  life,  and  also  for  the  developement  and  application  of  thoae  sub- 
jects ordinarily  treated  by  arithmetic,  the  pupil  is  referred  to  the  Author's  Pro* 
fressive  Practical,  and  Progressive  Higher  Arithmetics. 


186  PROMISCUOUS.  EXAMPLES. 

PROMISCUOW  EXAMPLES. 

1.  Multiply  the  difference  between  876042  and  8342GO 
by  176.  Ans.  7353632. 

2.  To  47320  add  three*  times  the  ^difference  between 
46270  and  31032.  Ans.  93034. 

3.  From  212462+432046,  take/rf?  7240— 230124. 

4.  Divide  the  sum  of  4802+560 10 +20342  by  4  times 
the  difference  between  1200  and  1082. 

Ans.  171ft*. 

5.  What  is  the  difference1  between   1824624+15624 
and  896042— 12342?  Ans.  956548. 

6.  What   is   the   difference   between   3426  x  284    and 
2001041  Ans  772880. 

7.  What  is  the  difference  between  3931476-^-556  and 
14x875?  Ans.  5179. 

8.  How  many  times«an  36  be  subtracted  from  1 1772  ? 

Ans.  327. 

9.  How  many  times  can  8  x  27  be  taken  from  1554768  ? 

10.  Divide  420x216  by  43756— 42851. 

Ans.   lOOyW 

11.  Multiply  3  times  the  sum  of^624+1036    by 
times  the  difference  of  375—296.  Ans.  2682840. 

12.  What  is  the  difference  between  5  times  2.5,  and 
5x2.5?  Ans.  11.25. 

13.  Multiply  4.05  +  .025+1.8  by  2—1.875. 

14.  Divide  5  by  .8  x  .025.    .  Ans.  250. 

15.  How  many  times  can  1.05  be  taken  from  4.725  ? 

Ans.  4.5  times. 

16.  To  .02  times  32.5  add  5.7  times  16.04—12.0026. 

Ans.  23166318. 


PROMISCUOUS   EXAMPLES.  187 

17.  What  is  the  difference  between  .675— .15  and  .23 
X.009?  Ana.  4.49793. 

18.  A  farmer  sold  a  horse  for  $140,  a  cow  for  $25,  and 
28  sheep  at  $2,50  a  head ;  how  much  more  did  he  receive 
for  the  horse  than  for  the  cow  and  sheep  ?       Ans.  $45. 

19.  A  young  lady  having  $75,  went  out  shopping,  and 
bought  14  yards  of  silk  for  a  dress,  at  $1,50  a  yard,  a 
shawl  for  $15,75,  a  bonnet  for  $8,  a  pair  of  gloves  for 
$1.125,  and  a  pair  of  shoes  for  $1,75;  how  much  money 
had  she  remaining  1  Ans.  $27.37|. 

20.  A  grocer  bought  12  firkins  of  butter,  each  contain- 
ing 56  pounds,  at  14  cents  a  pound ;  he  afterward    sold  5 
firkins,  at  16  cents,  and  7  firkins,  at  18  cents  a  pound ; 
how  much  was  his  whole  gain  1  Ans.  21.28. 

21.  A  miller  sold  256  barrels  of  flour,  at  $6.80  a  barrel, 
which  was  $475.60  more  than  the  wheat  from  which  it 
was  made,  cost  him  ;  what  was  the  cost  of  the  wheat  ? 

Ans.  $1265.20. 

22.  An  estate  worth  $25640,  has  demands  against  it  to 
*  the  amount  of  $9376 ;  after  these  claims  are  paid,  the 

remainder  is  to  be  divided  equally  among  5  individuals ; 
how  much  will  each  receive  ?  Ans.  $3252.80 

23.  If  15  tons  of  hay  cost  $311.70,  how  much  will  1 
ton  cost?  Ans.  $20.78. 

24.  Paid  $1.24  for  15.5  pounds  of  beef;  how  much  was 
the  price  per  pound  1  Ans.  $.08. 

25.  A  farmer  exchanged  21  bushels  of  wheat,  at  $2  a 
bushel,  for  cloth  worth  $3  a  yard  ;  how  many  yards  did 
he  receive  1  Ans.  14  yards. 

26.  A  man  having  labored  for  a  farmer  1  year,  at  $15 
a  month  expended  the  year's  wages  for  cows,  at  $18  each; 
how  many  cows  did  he  buy  ?  Ans.  10 


188  PROMISCUOUS   EXAMPLES. 

27.  What  will  be  the  cost  of  3  hogsheads  of  sugar,  each 
weighing  15  cwt.,.at  8  cents  a  pound  ?  Ans.  $360. 

28.  How  many  bushels  of  wheat,  at  $1.12  a  bushel,  can 
be  bought  for  $81.76  ?  Ans.  73. 

29.  If  140  barrels  of  apples  cost  $329,  how  much  is  the 
cost  per  barrel  ?  Ans.  $2.35. 

30.  At  $.825  per  bushel,  how  many  bushels  of  corn 
can  be  bought  for  $264?  Ans.  320. 

31.  If  25  yards  of  cloth  can  be  bought  for  $125.25,  how 
many  yards  can  be  bought  for  $751.50 1          Ans.  150. 

32.  If  150  bushels  of  wheat  cost  $435,  how  much  will 
311  bushels  cost?  Ans.  $901.90. 

33.  If  250  pounds  of  tea  cost  $135,  what  is  the  price 
per  pound  1  Ans.  $.54. 

34.  If  13  spoons  be  made  from  2  Ib.  10  oz.  9  pwt.  of 
silver,  what  will  be  the  weight  of  each  ? 

Ans.  2  oz.  13  pwt. 

35.  If  a  man  travels  20  mi.  3  fur.  36  rd.  in  a  day,  how" 
far  will  he  .travel  in  61  days  at  the  same  rate  ? 

Ans.  1249  mi.  5  fur.  36  rd. 

36.  If  I  put  376  gal.  3  qt.  1  pt.  of  cider  into  9  equal 
casks,  how  much  do  I  put  into  each  cask  1 

37.  If  a  family  use  1}  pounds  of  tea  in  1  month,  how 
much  would  they  use  in  1  year  ?  Ans.  13i  pounds. 

38.  What  would  be  the  cost  of  565  pounds  of  butter 
at  12i  cents  a  pound?  Ans.  $70.625. 

39.  At  $4.25  per  bushel  how  much  clover-seed  can  be 
bought  for  $11.6875  1  Ans.  2.75  bushels. 

40.  At  -Jg-  of  a  dollar  a  pound,  what  will  be  the  cost 
of  12  pounds  of  sugar  ?  Ans.  $.75. 

41.  At  |  of  a  dollar  a  yard,  what  will  be  the  cost  of 
40|  yards  of  cloth?  Ans.  $15.30. 


PROMISCUOUS   EXAMPLES.  189 

42.  How  many  cubic  yards  of  earth  must  be  thrown 
from  a  cellar  40  ft.  long,  30  ft.  wide,  6  ft.  deep;  and  what 
will  be  the  cost  of  the  excavation,  at  12^  cents  a  cubic 
yard  ?  Ans.  2662-  cubic  yards ;  $33.33i. 

43.  If  6  pounds  of  cheese  cost  $£,  how  much  will  10 
pounds  cost?  Ans.  $li. 

44.  HDW  much  wheat  at  SI. 25  a  bushel,  must  be  given 
for  50  bushels  of  corn  at  $.70  a  bushel  1 

45.  At  10  cents  a  pint,  how  much  will  189  gallons  of 
molasses  cost?  Ans.  $151.20. 

46.  At  15  cents  a  pound,  how  much  will  -fa  of  a  pound 
of  coffee  cost  ?  Ans.  2f  cents. 

47.  If  3  gallons  of  molasses  cost  $£ ,  how  many  gal- 
lons can  be  bought  for  $4  1  Ans.  14f . 

48.  At  $7j  a  firkin,  how  many  firkins  of  butter  can  be 
bought  for  $33  ?  Ans.  4f . 

49.  If  £  of  a  yard  of  cloth  cost  $4,  what  will  one  yard 
cost "?  Ans.  $24. 

50.  At  $3  a  barrel,  how  many  barrels  of  cider  can  be 
bought  for  $8£  ?  Ans.  2}|  barrels. 

51.  What  part  of  100  pounds  is  16  pounds? 

Ans.  -^g. 

52.  How  much  wood  in  a  load  10  ft.  long,  3|  ft.  wide* 
and  4  ft.  high  ?  Ans.  1  Cd.  12  cu.  ft. 

53.  How  many  tons  of  coal  may  be  bought  for  $346.125 
at  $9.75  per  ton  ?  .       Ans.  35.5  tons. 

54.  What  is  the  interest  on  $136.80  for  1  yr.  11  mo., 
at  7  per  cent.?  Ans.  $18.354. 

55.  What  will  be  the  cost  of  .6  of  a  gallon  of  wine,  at 
$.65  a  gallon  1  Ans.  $.39. 

56.  A  owns  4  of  a  flouring  mill,  and  sells,  f  of  his  shard 
to  B  ;  what  part  of  the  whole  has  he  left  ? 


190  PROMISCUOUS    EXAMPLES. 

57.  If  2  yards  of  cloth  cost  $6f ,  how  much  will  9  yards 
cost?  .  Ans.  $30£. 

58.  What  will  |  of  |  of  a  barrel   of  flour   cost  at  $7^ 
per  barrel1?  Ans.  $2|. 

59.  If  1   acre  of  land   yield   1  T.  9  cwt.  1  qr.  22  Ib.  oi 
hay,  how  much  will  18  acres  yield  ? 

GO.  A  speculator  bought  1575  barrels  of  potatoes,  and 
upon  opening  them,  he  found  15  per  cent.of  them  spoiled; 
how  many  barrels  did  he  lose  ?  Ans.  236.25. 

61.  How  many  steps  of  30  inches   each,  must  a  person 
take  in  walking  10  miles?  Ans.  21120. 

62.  A  man  bought  12  bushels  of  chestnuts,  at  $4.50  a 
bushel,  and  sold  them  at  12  cents   a  pint ;  how  much  was 
his  whole  gain?  Ans.  $88.16. 

63.  What  is  the  interest  on   $300,  for  10  mo.  21  days, 
at  6  per  cent.  ?  Ans.  $16.05. 

64.  An   agent   in  Chicago,   purchased    5450  bushels  of 
wheat,  at  $.82  a  bushel ;  what  was   his   commission  at  2 
per  cent,  on  the  purchase  money  1  Ans.  $89.38. 

65.  A  vessel    loaded  with   4500  bushels  of  corn,  was 
overtaken  by  a  storm  at  sea,  and  it  was  found   necessary 
to  throw  overboard  25  per  cent,  of  her  cargo ;   what  was 
tfie  whole  loss,  at  60  cents  a  bushel  ?  Ans.  $675. 

66.  A  grocer  bought  2  hogsheads  of  molasses,  at  37^ 
cents  a  gallon,  and  sold    it  at  20  per  cent,  advance  on  the 
cost ;  how  much  was  his  whole  gain?  Ans.  $9.45. 

67.  If  f    of  acres  of  land   is  worth   $60,  what   is   the 
value  of   1  acre?  Ans.  $84. 

68.  If  ly  bushels  of  wheat   sow  an   acre  of  land,  how 
many  acres  will  12  bushels  sow?  Ans.  9  acres. 

69.  If  a  farm    is   worth   $3840,  how   much   is  f  of  it 
worth  ?  Ans.  $2400. 


PROMISCUOUS    EXAMPLES.  19] 

70.  If  17  kegs  of  nails   weigh   27  cwt,  3  qrs.  23  Ibs. 
3oz.,  long  ton  weight,  how  much  will  1  keg  weigh1? 

71.  If  a  bushel  of  apples  cost  §  of  a  dollar,  how  many 
may  be  bought  for  f  of  a  dollar  ? 

72.  Divide  £  of  f  by  j  of  |  ?  Ans.  J. 

73.  What  is  the  amount  of  $620  for  4   yr.  3  mo., 
t  6  per  cent.  ?  Ans.  $778.10. 

74.  What   is  the  brokerage  on   $5462,  at  4  per  cent., 

75.  How  many  pounds  of  butter  at  13|  cents  a  pound, 
must   be   given  for    1230   pounds  of  sugar   at   8   cents  a 
pound?  Ans.  728f  pounds. 

76.  Divide  168  bu.  1  pk.  6  qt.  of  corn   equally  among 
35  persons.  Ans.  4  bu.  3  pk.  2  qt. 

77.  What  will    be   the   cost  of  lathing   and  plastering 
overhead,  a   room   36  feet   long  and  27  feet  wide,  at   28 
cents  a  square  yard  1  Ans.  $30.24. 

78.  How  much  land  at  $2.50  an  acre,  must  be  given  in 
exchange  for  360  acres,  at  $3.75  an  acre  ? 

79.  What  is  the  amount  of  $564.58,  for  3  yr.  5  mo. 
12  da.,  at  6  per  cent  ?  Ans.  $681.448. 

80.  How  much   sugar   at  9  cents   a   pound,  should   be 
given  for  6^  cwt.  of  tobacco,  at  14  cents  a  pound  1 

81.  How   many  times   may  a  jug  which   holds   J  of  a 
gallon, be  filled  from  a  cask  containing  128  gallons'? 

82.  A  man  having  $25000,  invested  30  per  cent,  of  it  in 
bonds  and  mortgages,  45   per  cent,  of  it   in  bank  stocks, 
and  the  remainder  in  railroad  stock;    how  much   did  he 
invest  in  railroad  stock  ? 

Ans.  $6250. 

83.  How  many  times  can  a  box  holding  4  bu.   3  pk 
2  qt,  be  filled  from  336  bu.  3  pk.  4  qt.  ? 

Ans.  70. 


192  PKOMISCUOUS    EXAMPLES. 

84.  How  many  cords  of  wood  in  17  piles,  each  11  feet 
long,  4  feet  wide,  and  6  feet  high  1 

85.  If  the  price  of  1  acre  of  land  is  $32f ,  what  is  the 
value  of  |  of  an  acre  ?  Ans.  $28f  |-. 

86.  What  number  of  times  will   a  wheel  14  ft.  10  in. 
in  circumference,  turn  round  in  traveling  11  mi.  6  fur 
15rd.  12ft.  6  in.?  Ans.  4200. 

87.  A  man  bought  a  farm  of  136  acres,  at  $94  an  acre ; 
he   paid   $475   for  fencing  and  the   improvements,  and 
then  sold  it  for  14  per  cent,  advance  on  the  whole  cost ; 
how  much  was  his  whole  gain  ? 

Ans.  $1856.26. 

88.  If  36.48  yards  of  cloth  cost  $54.72,  how  much  will 
14.25  yards  cost  ?  Ans.  $21.375. 

80.  If  $13.342  will   pay  for    17.5    bushels  of  barley, 
how  many  bushels  can  be  bought  for  $76.24 1 

Ans.  100  bushels. 

90.  A  lady  having  $40.50,  spent  40  per  cent,  of  it  for 
dry  goods;  how  much  had  she  left? 

Ans.  $24.3b. 

91.  A  gentleman  bought  a  house  and  lot  for  $6425  •  in 
the  course  of  five  years  it  increased  in  value  110  per  cent, 
how  much  was  the  property  then  worth  1 

Ans.  813492.50. 

92.  How  much  will   a   broker   charge   to  chahge  $560 
uncurrent  money  for  currehTmt)lfey;"^^3^)cr^ee«t.  1 

Ans.  $16.80. 

93.  If  4  hogsheads  of  wine  cost  $181.44,  what  will  be 
the  cost  of  1  pint  ?  Ans.  9  cents. 

94.  What  will  5  casks   of  rise  cost,  each  weighing  165 
pounds,  at  3d  per  pound,  Georgia  currency  ? 

Ans.  $44.198- 


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